Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 4 Probability Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $Y=$ the number of people you have known personally that have committed suicide uithin the past 12 months According to recent Gencral Social Surveys, the piobabilities for the potential values of $Y$ are approximately $P(0)=90, P(1)=.08, P(2)=.02$.
a) Conslruct a probability disuibution and its bistogram for $Y$
b) Find the mean of the distribution of $Y$.

Sriparna Bhattacharjee

Numerade Educator

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Problem 2

Let $Y=$ number of people you lave known personally uho came doun with AIDS Supmose Table 3.9 refers to the population of inlerest
a) Construct the probability distribution for $\gamma$
b) Find the mcan of the distnbution of $Y$.

Victor Salazar

Numerade Educator

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Problem 3

Five students, Ann. Betty. Clint. Douglas, and Edu arrd. are rated equally qualificd for admission to law school. ancad of othel applicants However. all but two positions have been filled for the entenng class Since the admissions cominituee can admit only ruo more students, it decides to randomly select two of these five candidates. For this strategy, Ici $Y=$ number ol fenales admitted. Using the first letter of the name to denote a student, the different combinations that could be admitted are (A. B), (A, C), (A. D), (A. E). (B. C). (B, D), (B. E), (C. D), (C, E), and (D. E)
a) Construcs the probability distribution for $Y$.
b) Find the expected value of $Y$

Victor Salazar

Numerade Educator

00:47

Problem 4

Refer to the previous execise. Construct the probability distibution of $\chi=$ number of fcunales adnitted. when three of the five students are randonly selected for admission. Find the mean of the distribution

Hoan Nguyen

Numerade Educator

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Problem 5

Let $Y$ be the outcome of selecting a single digit from a random number table.
a) List the possible values for $Y$
b) Construct the probabilit) distnbation for ) (This typc of distnbution is called a uniform distnbution. because of the uniform sprcad of probabilities ucross the possible outcomes.)
c) Find the mean of this distribution
d) The standard deviation $\sigma$ of this distrbution is one of the follouing 07.2 .9509 .0 . Pich the correct value, and justrify.

Ana Carolina Da Cruz

Numerade Educator

03:54

Problem 6

In a statewide lottery, one can buy a ticket for $$\$ 1$$. With poobability .0000001. one wins a million dollars (S1.000,000), and with probability 9999999 one vius nothing ( $$\$ 0$$ ).
a) Let $Y$ denote the uinnings from luying one ticket. Construct the probability distribution for $Y$. Show that the mean of the distribution cquals 10 , corresponding to an expected return of 10 cents for the dollar paid.
b) The pront $X$ from buy ing a single $$\$ 1$$ tickel equals the winnings $Y$ from playing the lotlery minus the dollar paid for the ticket, that is. $X=Y-1$. Construct the probability distribution of $X$. and find its mean. Inlerpret

Joshua Argo

Numerade Educator

02:18

Problem 7

For a nomal distribution. find the probability that an observation is
a) At least one standard deviation above the mean.
b) At least oue standard deviation below the mean.
c) At least . 67 slandaid devidtions above the mean
d) At least 2.33 standard deviations above the mean.

Lynn Larson

Numerade Educator

03:39

Problem 8

For a normally distributed variablc. venly that the ptobability between
a) $\mu-\sigma$ and $\mu-\sigma$ equals 68
b) $\mu-1.96 \sigma$ and $\mu+1.96 \sigma$ equals .95 .
c) $\mu-3 \sigma$ and $\mu-3 \sigma$ equals 997 .
d) $\mu-.67 \sigma$ and $\mu+.6 \bar{\imath} \sigma$ equals .50

Sheryl Ezze

Numerade Educator

02:52

Problem 9

Find the $z$-value for which the probability that a normal variable exceeds $\mu+z \sigma$ cquals
a) .01
b) . 025
c) .05
d). 10
e) .25
f) . 50

Sanchit Jain

Numerade Educator

01:50

Problem 10

Find the $z$-yalue such that the mterval from $\mu-z \sigma$ to $\mu+z \sigma$ contains a) $30 \%$. b) $90 \%$, c) $95 \%$, d) $98 \%$, e) $99 \%$ of the probability for a nonnal distribution.

Sanchit Jain

Numerade Educator

01:30

Problem 11

Find the $\tau$-values corresponding to the a) 90 th$, b) 95 th$, c) 98 th , and d) 99 th pescentiles of a nornal distribution

James Kiss

Numerade Educator

01:58

Problem 12

Sbow that if $z$ is the number such that the interval from $\mu-z \sigma$ to $\mu+z \sigma$ contains $90 \%$ of a normal distribution, then $\mu+z \sigma$ equals the 95 th percentile.

Wendi Zhao

Numerade Educator

04:52

Problem 13

If $z$ is the positive number such that the iuterval from $\mu-\Sigma \sigma$ to $\mu+; \sigma$ contans $50 \%$ of a nonnal distribution, then which percentile is a) $\mu+z \sigma$ b) $\mu-\tau \sigma$ ? Find this value of $z$. Lsing this result, explain how to find the upper quartile, lower quartile, and interquartile tange of a nommal distribution,

Maxime Rossetti

Numerade Educator

01:38

Problem 14

An observation is .50 standard deviations below the mean on a normally distributed variable. What ptoportion of the data fall below that observation? Above it?

Joshua Sieverding

Numerade Educator

02:46

Problem 15

What proportion of a normal distnbution falls in the following ranges?
a) Above a z-score of 2,10 .
b) Below a $\bar{z}$-score of -210 .
c) Above a 2 -score of $-2,10$.
d) Bctween :-5cores of $-2,10$ and 210 .

Melissa A

Numerade Educator

01:00

Problem 16

Find the $z$-score for the number that is less than ouly 1 \% of the values of a nonnal distnbution

Linh Vu

Numerade Educator

02:21

Problem 17

Find the $z$-scores for the lower and upper quartiles of a normal distribution.

Melissa A

Numerade Educator

02:26

Problem 18

According to Current Population Reports, self-employed individuals in the L'nited Slates in 1990 worked an dverage of 44.6 hours per week. with a standard deviation of 14.5. Assunling this variable is approximately normally distributed, tind the proportion who averaged inore than 40 hours per week.

Yingtai Xiao

Numerade Educator

02:23

Problem 19

The Mental Development Index (MDI) of the Bayley Scales of Infant Developynent is a standardized incasure used in longitudinal follow-up of high-rish infants. It hass approximalely a normal distnbution with a mean of 100 and a standard deviation of 16 .
a) What proportion of children have a MDI of at least 120 ?
b) What proportion of children have a MDI of at least 80 ?
c) Find the MD1 score that is the 90 th percent土e
d) Find the MDI score such that $10 \%$ of the population have MDI scores below that value.
e) Find and interpret the lower quarule, median, and upper quartilc of MDJ
f) Find the interquartile range of MD1 scores, and, using the definition of outlicr in terms of the $1 Q R$, find the ranges of MDI scores that would be considered outhers

Sheryl Ezze

Numerade Educator

02:48

Problem 20

For 5459 pregnant women using Aarhus University Hospital in Denmurk in a two-year period who reported infornation on length of gestation until bith, the mean was 2819 days. with standard deviation 114 days. A baby is classified as premature if the gestation ume is 258 days or less (Bivtish Medical Journal. Vol 307, July 24, 1993, p. 234).
a) If gestation times aue nornally distribured, what proportion of babies would be horn prematurcly?
b) The actual proportion bon! prematurely during this period was 036 Based oli this information, how would you expect the distribution of gestauon time to differ from normal ${ }^7$ ?

Sheryl Ezze

Numerade Educator

02:54

Problem 21

Suppose that the weekly use of gasolinc for motor travel by adults in North Amenca is approximate.) nomally distributed, with a mean of 16 gallons and a standard deviatron of 5 gullons.
a) What proportion of adults use more than 20 gallons per week?
b) Find and interpret the low er and upper quartules and interquartile range of gasoline use. c) Assuming that the standard deviation and the normal form would remain constant, to what level must the mcan reduce so that only $5 \%$ use more than 20 gallons per week?
d) If the disu ibution of gasoline use is not actually nonnal. how would you expect it to deviate from normal?

Nick Johnson

Numerade Educator

02:56

Problem 22

In 1992, the murder rates (per 100.000 residents) for 74 majos U.S. citics had a mean of 20.8 and a standard deviation of 14.6 (Stalstical Abstract of the United Slates. 1994)
a) Could the distribution of murder rates be well upproximated by a normal distribution? Why or why not?
b) One of these cities, Washington, D.C . had a murder rate of 75.2 . Find its $z$-scorc. If the distributon were ıoughly normally distributed. would the be an unusually high value? Why?
c) What is the murder rate of a city that has a $z$-score of 0.0 ?

Sheryl Ezze

Numerade Educator

Problem 23

Suppose that monthly rental rates for one-bedroom unfurnshed apartments in Ann Arbor, Michigan. are approximately normal in distribution, with a mean of $$\$ 750$$ and a standard dcviation of $$\$ 150$$ a) What proportion of the rental rates are at least $$\$ 1000$$ per montll?
b) What propos tion of rates are less than $$\$ 600$$ per month?
c) What proportion are berween $$\$ 500$$ and $$\$ 1000$$ per month?

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Problem 24

Suppose that pioperty taxes on bomes in Gainesville. Florida, are approximately nonnal in distribution, with a mean of $$\$ 1400$$ and a standard deviation of $$\$ 60_0$$ The property tax for one particular hone is $$\$ 1700$$.
b) What proportiou of the property laxes excced $$\$ 1700$$ ?
e) In fact. the true distrbution is moundslaped but not normal. Hou would you expest it to deviate from nomaal?

Kari Hasz

Numerade Educator

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Problem 25

An encigy study in Gainesvillc. Flot idal. 1ound that in a n pical month household usc of electricity has a mean of 800 and a standard der iation of 500 (kilowatt-bouns). Each ohsen auon is converted to a $z$-५core. Each hensehold with a $$1.0$$ (i.e.. electncity use more than one standard deviation above the inean) is sent a notice suggesting a reducrion in electical use, for conscrvation purposes.
a) If the distribution were normal, what pescentage of the houscholds would teceive this notice?
b) Do you think the distribution is tuly nommal" Why or why not?

Rashmi Sinha

Numerade Educator

02:58

Problem 26

Refer to Problem 4.3 For 1andomnly selecting tuo of the five students, construct the sampling distribution of the proporion of the students selected who are female What is the probability that the proportion equals 0 ?

Sarah X

Numerade Educator

04:10

Problem 27

Construct the sampling disuribution of the proportion of heads. for liipping a balanced coin
a) Once
b) Twice.
c) Three times
d) Four times
e) Describe how the shape of the sanpling distribution seems to be changing as the number of flips increalses.

Nick Johnson

Numerade Educator

02:18

Problem 28

The pubability disurbution associated with the outoone of relling a balanced die has probabifity $1 / 6$ attacled to each integer, $11,2.3$ 4.5.6\}. Considen the experiment of rolling a balanced die twice
a) Enumesale the 36 possible pairs of numbers, takng into account the order of rolling (e g $(1,2)$, representing a 1 followed by a 2 . is difficrent fiom $(2,1)$.
bj Treating these 36 ןaairs as equally lilely, construct the sampling distribution for the sample mean of the two nuinoers rulled.
c) Construct a histogranu of the sampling distribution in (b). What is its shape?
d) Calculate the mean of the sampling distmbution in (h), and slow that it is identical to the mean of the population distribution.

Sheryl Ezze

Numerade Educator

04:30

Problem 29

According to Current Popularion Reports, the population distribution of number of yeans of education for self-employed individuals in the United States in 1990 l.ad a mean of 13.6 and a standard deriation of 30 Find vhe mean and ctandard des iation of the sampling distriburion of $\bar{r}$ for a randomn sample of a) 9 residente b) 36 residents. c) 100 Iesidents Describe the pattem as $n$ increases.

Rashmi Sinha

Numerade Educator

01:01

Problem 30

Refer to Problem 4.6. The mean and standard dev iation of the probability distribution for the lottery winnings $Y$ are $\mu=10$ and $\sigma=316.23$. Suppose you play the lottery 1 million times Let $\bar{Y}$ dcnotc your average winnings.
a) Find the mean and standard deviation of the distribution of $\bar{Y}$.
b) How likely is it that you would "come oul ahead." with your average uinnings exceeding $$\$ 1$$. the amount you paid to play cach time?

Hossam Mohamed

Numerade Educator

02:55

Problem 31

Suppose that neekly incomes of imgrant workers in California have a mean of $$\$ 250$$ and a standard deviation of $$\$ 75$$.
a) If the distribution is normal. find the probability that a mugrant worker has incone oves $$\$ 300$$.
b) For a random sample of nine migrant workers. report the sanspling distribution of the sample mean income. Find the probability that the sample mean exceeds $$\$ 300$$.

Jon Southam

Numerade Educator

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Problem 32

According to recent General Social Surveys, in the United Stales the distribution of $Y=$ number of sex partners you have bad in the past 12 months has a mean of about 1.0 and a standad deviation of about 1.0 .
a) Does $Y$ have a nommal distribution? Explain.
b) For a random sample of 100 addles. find the sampling distribution of $\bar{y}$
c) Rcfer to b). Report an interval within which the sample mean would almost surely fall.

Danielle Fairburn

Numerade Educator

01:49

Problem 33

According to Siatistical Abstract of the United States, 199.5, houschold stzc has a mean of 2.6 and a slandard deviation of 15 . For a random sample of 225 homes. find the probability that the sample mean household size falls within. 1 of the population mean

Sheryl Ezze

Numerade Educator

Problem 34

The distribution of famuly size in a particular trbal society is skewed to the right., with $\mu=5.2$ and $\sigma=3.0$. These values are unknown to an anthropologist, $u$ ho takes a sample of families in this society in order to estimate mean family size. Let $\bar{Y}$ denote the sample mean fanily size she obtains, for a random sample of 36 famulies.
a) Specify the sanpling distribution of $\bar{Y}$.
b) Find the probability that her samplc mean falls within .5 of the true mean
c) Suppose she takes a randons sample of size 100 . Find the probability that the sample mean falls within 5 of the truc mean, and compare the answer to that obtained in part b) d) Refer to part c). If the sample were tuly random, would you be surprised if the anthropologist obtained $\bar{Y}=4.0$ ? Why? (This could well leappen if the sample werc not randoni.)

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02:23

Problem 35

The scores on the Psychotnotor Devclopment Index (PDI), a scale of infant development, arc approximately nonnally distributed with mean 100 and standard deviation 15. An infant is selecled at random.
a) Find the probability that the infant's PDI score is al least 100
b) Find the probability that PDI exceeds 103 .
c) Find the probability that PDI is between 97 and 103
d) Find Uie $z$-score for a PDJ value of 90 . Would you be surprised to observe a value of 90 ?
e) Suppose we convert all the PDI observations to $=$-scores. that is. for each infant. subtract 100 from the value of PDI and divide by 15 Then, uhat is the distribution of the $i$-scores called? What are the mean and standard deviation of these $z$-scores?

Sheryl Ezze

Numerade Educator

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Problem 36

Refel to the previous cxercise. A study uses a random sample of 225 infants.
a) Describe the sampling distribution of the sample mean PDI.
b) Find the probability that the sample mean exceeds 103 .
c) Find the probability that the sample mean falls between 97 and 103
d) Find the $z$-score from the sampling distribution corresponding to a sample mean of $\bar{Y}=90$, when the sample size is 225 . Would you be surprised to observe a sample mean PDI of $90^2$ Why?
e) Compare the results of parts (a)-(d) with those in the previous exercise, and interpret the differences.
f) Repeat parts (a)-(d) for a random sample size of $n=25$ and compare the results to those for $n=225$

Shu Naito

Numerade Educator

02:53

Problem 37

Refer to the previous two exercises
a) Sketch the population distribution for PDI
b) Superimpose, on the sketch of part (a), a sketch of the sampling distribution of the sample mean, for a random sample of 225 intants
c) Supenmpose, on the above sketches. a sketch of the sampling distribution of the sample meall for a random sample of 25 infants.

Ryan Mcalister

Numerade Educator

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Problem 38

We need an estimate of the incan acreage of Canadian fanns. We plan to measure acreage for a random sample of 100 farms. Resuds from an earlier study suggest that 200 acres is a reasonablc guess for the standard deviation of farm size. Find the probability that the sample mean acreage falls within 10 acrcs of the population mean acreage.

Danielle Fairburn

Numerade Educator

03:37

Problem 39

An executive in a savings and loan association decided to estimatc the inean amount of money lent to individuals for financing higber education in the past year From past experrence, she believes that $$\$ 240$$ is a reasonable guess for the standard deviation of the distribution of loan amounts. She would like her estimate of the mean to be within $$\$ 50$$ of the actual mean. Find the probability that this happens under the following conditions.
a) She takes a random sumple of 36 loan records.
b) She takes a random sample of 64 loan records
c) She takes a random sample of 144 loan records.

Sheryl Ezze

Numerade Educator

04:44

Problem 40

A study investigating the relationships among votung patterns, pohtical opinions, and age takes a random sample of 100 undividuals in a typical precincl.
a) If the standard deviation of the ages of all individuals in the precinct is $\sigma=15$. find the probability that the mcan age of the individuals sampled is within 2 years of the mean agc for all individuals in the precinct.
b) Would the probability be larger, or smaller, if $\sigma=10$ ?

Kayla Laughman

Numerade Educator

04:38

Problem 41

Refer to Problem 1.7. Using the population defined by your class or using the WWW data set, the instructor will select a variable, such as weekly time watching telerision
a) Construct a histograsu or stem and leaf plot of the population distriburion of the variable for the class.
b) Using a random number table, each student should select five srudents at random and compute the sample mean response for those students. (Each student should use different random numbers.) Plot a histogram of the sample nieans oblained by all the students. How do the spread and shape compare to the histogram in ta)? What does this illustrate ?

Jerrah Biggerstaff

Numerade Educator

02:53

Problem 42

Table 4.4 provides the agcs of all 50 heads of households in a sunall Nova Scotian fishmg village. The distribution of these ages is characterized by $\mu t=47.18$ and $\sigma=14.74$.
a) Consurect a stein and leaf plot of the population distrbution of the ages of all heads of houscholds
b) Using a random number table, each student should select nine random numbers betheen 01 and 50 . Using these numbers. each student should sample nine heads of households and compute their sample mean age. Using the intervals 31.01-34.00. 34.01-37.00, $3701-40.00$, and so forth to $61.01-64.00$, construct the empirical sampling distribution of the $\bar{Y}$-values Compare is to the distribution in parl (a)
c) Find the mean of the $\bar{Y}$-values generated in part (b) What value do you expcct for this mican in a long run of repeated samples of size 9 ?
d) Find the standard deviation of the $\bar{Y}$-valucs gene atod in part (b). What value do you expect for this standard deviation in a long run of repealed sumples of size 9 ?

Ryan Mcalister

Numerade Educator

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Problem 43

For a single toss of a coin, let $Y=1$ for a head and $Y=0$ for a tail
a) Assuining the coin is balanced, construct the probabulity disuibution for $Y$. and calculate its meall
b) The coin is flipped ten times. yielding six heads and four tails. Consirucl the sample relatuve frequency disinbution
c) Each student in the class should Ilip a coin ten times and calculate the proportion of heads in the sample. Summarize the empirical sampling distribution by constructing a histogran or stem and leaf plot of the ptoportions obtamed Describe the shape and spread of the sanyling disuibution compared to the distributions in (a) and (b). Coinpute the mean and standard deviation of the sample proportion values.
d) If we performed the exper iment in (c) an indefintely lange number of tincs, what values would we get for the (i) mean and (ii) standatd deliation of the sample proportion values?

Jason Gerber

Numerade Educator

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Problem 44

The previous exercise assumcs the coin is balanced. Suppose instcad that the probability of a head equals 75 and the probability of a tail equals . 25
a) Construct the probability disrribution of $Y$ the outcome of a single flip. and calculate its mean.
b) Each student in the class should simulate the results of flipping this coin ten times hy seleclins ten two-digit numbers Irom the random number table; a number between 0 of and 74 represents a bead and a nuinber between 75 and 99 represents a tail Again, construct a plot of the sample proportions. which is an empincal approximation for the true sampling distribution. Computc the mean and standard deviation of the sample proportion values Report the theoretical value for the mean of the samphng distribution. (From Problem 4.55(c), the theoretical standard deviation is .J37)

Jason Gerber

Numerade Educator

06:12

Problem 45

Each student should bring ten coins to class The observation for each coin is its age. the difference between the current year and the year on the coin.
a) Using all the students' observations, the class should construct a stem and lear plot of the ager. What is its shape?
b) Now each student should compure the mean for that student's ten coins, and the class should construct a plot of the means. Wrat type of distribution is this, and how does it compare to the one in (a)?
c) What concepts does this exercie illustrate?

Khalida Dawar

Numerade Educator

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Problem 46

a) Which disinbution doce the sample distnbution tend to resernble more closely-the sampling distrubution or the population distrbution? Explain.
b) Explain carefully the difference hetween a distribution of sample measurements and the sompling distribution of $\bar{Y}$. Illustrate your answer for a variable $Y$ that can only take values of 0 and 1 .
Select the correct response(c) in the following multiple-choice questions.

Shu Naito

Numerade Educator

Problem 47

The standard error of a statislic describes
a) The standard deviatoon of the sampling discribution of that statistic.
b) The standard deviation of the sample measurements.
c) How close that statistic is likely to fall to the parameter that it esumates.
d) The varability in the valucs of the statistic for repeated random samples of size $n$.
e) The error that occurs due to nonresponse and measurement errors.

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01:15

Problem 48

The Central Limit Theorem implies that
a) All variables have approximately bell-shaped sample distributions if a random sampie conlains at least 30 olsecrvations.
b) Population disuibutions arc nonnal whenever the population size is large.
c) For large random samples. the sampling distribotion of $\bar{Y}$ is approximately normal. regardless of the shape of the population distribution.
d) The sampling distribution looks more like the popalation distribution as the sainple size increases
e) All of the above.

Jerrah Biggerstaff

Numerade Educator

01:31

Problem 49

The numbers in the body of Tablc A provide tail probabilities
a) For the normal distribution.
b) Fol any symmetric distribution.
c) For uny distribution
d) Only for the standard normal distribution

Tyler Moulton

Numerade Educator

00:56

Problem 50

True or Falsc: As the sample size increases. the standad error of the saingling distribution of $\bar{Y}$ increases.

Neel Faucher

Numerade Educator

08:12

Problem 51

Lake Wobegon Junior Collcge adinits students only if they score above 400 on a standardized achievement lest Applicants from group $A$ have a incan of 500 and a standard deviation of 100 on this test. and applicants from group $B$ have a mean of 450 and a standard deviation of 100 Both distributions are approximalely normal, and both groups have the same size
a) Find the proportion not admitted for each group.
b) OI the students who are not admitted, what proportion ate fion group $B$ ?
c) A state legislator proposes that the college lower the cuioff point for adinission to 300 , thinking that the proportion of the students who are not admitled who are from group $B$ would decrease If this policy is implemented, deternine the effect on the answer to (b). and commeit.

Nick Johnson

Numerade Educator

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Problem 52

For a normal distribution, show thal
a) The upper quartule equals $\mu+67 \sigma$.
b) The interquartile range, $1 Q R$, is related to $\sigma$ by $I Q R=1.35 \sigma$.
c) For sample data thal are approximately nomal, an outlier (according to its definition for box plots) is an obser ation falling more than 2.7 standard deviations below or above the mean, and this happens for only $07 \%$. of the data.

Jason Gerber

Numerade Educator

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Problem 53

Sunshine City was designed to aluract retired people. and its current poputation of 50.000 residents has a mean age of 60 years and a standard deviation of 16 years. The distrubution of ages is skcwed to the leit. icflectug the predomnitance of older individuals A randon sample of 100 residents of Sunshine City las $\bar{Y}=58.3$ and $s=15.0$.
a) Describe the population disiribution.
b) Describe the sample distribution, What shape does ia probably have?
c) Describe the sampling disinbution of $\bar{Y}$ for $n=100$. What shape does it have?
d) Describe the sampling distribution of $\bar{Y}$ for a sample of size $n=1$
e) Describe the sampling distribution of $\bar{Y}$ for a sample of size 50.000
f) Suppose random samples of size $n=100$ were ıepeatedly selected. so that each resident has an equal chance of being closen for any onc obser ation in tach sample of size 100 Describe the likely appeal ance of the histogram of $\bar{Y}$ values
g) Would it be higtly unusual to observe a subject of age under 40 in Sunshine City? Would in he highly unusual to observe a sample mean under 40 . for a sample size of 100 ? Explain.

Sheryl Ezze

Numerade Educator

02:46

Problem 54

Refer to Example 4.7 Compute the standard ertor when $\pi=10,100,1000$ and 10,000 . In each case. use the Empirical Rule to predict an interval within which the sample pioportion is almost certain to fall. Notice that the interyal shrinks in width as the sample size increases. This is a conscquence of the law of large munbers, which states that the sample proportion tends to eventually converge to the population proportion as the sample size incrcases indefinutely.

Sheryl Ezze

Numerade Educator

27:31

Problem 55

The vanance of a discrete prohability distribution is
$$
\sigma^2=\sum(y-\mu)^2 P(y)
$$
the average squaned deviation. This also cquals
$$
\sigma^2=\left[\sum y^2 P(x)\right]-\mu^2
$$
a) Suppose $Y=1$ with probability .5 and $Y=0$ with probability 5, such as in Example 4.7 on preference for Democratic and Requblican candidate Lsing each fomula show that $\sigma^2=.25$. and hence the standard deviation $\sigma=.50$.
b) Suppose $Y=1$ with probiability $\pi$ and $Y=0$ with probability $1-\pi$, where $\pi$ denotes a fixed number belween 0 and 1 . Show that $\mu=\pi$ and that $\sigma^2=\pi(1-\pi)$
c) Refer to (b) Show that the standard error of a sample proportion for a random sample of size $n$ eqtals $\sqrt{\pi(1-\pi) / n}$.

Jeremiah Mbaria

Numerade Educator

03:04

Problem 56

Refer to the fonnula for the variance of a probability distrbution in the previous cx crcise. Find the standard deviation fot the distnbution in Probtem 4.1. Can one use the Empinval Rulc to interpret this slandard deviation? Explain.

Christopher Stanley

Numerade Educator

01:13

Problem 57

The curve for a nonnal distribution with mean $\mu$ and standard deviation $\sigma$ has mathematical fonnula
$$
f(1)=\frac{1}{\sqrt{2 \pi} \sigma} e^{-(1-\mu)^{\prime} /\left(\pi \pi^2\right)}
$$
(The integral of this function with respecf to 1 between $\mu+\Sigma \sigma$ and $\infty$ equals the tail probability tabulated in Table A). Show that this curve is symmetric: that is. for any constant $c$. the curve has the same value for $v=\mu+c$ as for $v=\mu-c$.

Carson Merrill

Numerade Educator

01:06

Problem 58

The standard error formula $\sigma_5=\sigma / \sqrt{n}$ actually treats the population size $N$ as infintitely large relative to the cample size $n$. The formula tor $\sigma_{\dot{p}}$ for a finite population size $N^T$ is
$$
\sigma_{\dot{r}}=\sqrt{\frac{N-n}{N-1}}\left(\frac{\sigma}{\sqrt{n}}\right)
$$

The term $\sqrt{(N-n) /(N-1)}$ is called the finite population correction.
a) When $n=300$ students are selected from a college student body of $\operatorname{sizc} N=30,000$, show that $\sigma_{\hat{\gamma}}=.995 \sigma / \sqrt{n}$. (In practice, $n$ is usually small relative to $A_1$, so the correction has little influence.)
b) If $n=N$ (i.e, we sample the entire population), show that $\sigma_{\hat{y}}=0$. In other words, no sampling error occurs. since $\bar{Y}=\mu$.

Sheryl Ezze

Numerade Educator