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Essentials of Statistics for the Behavioral Science

Frederick J Gravetter, Larry B. Wallnau, Jon-David Hague

Chapter 7

Probability and Samples: The Distribution of Sample Means - all with Video Answers

Educators

Chapter Questions

03:21

Problem 1

Briefly define each of the following:
a. Distribution of sample means
b. EXpected value of $M$
c. Standard error of $M$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:27

Problem 2

Describe the distribution of sample means (shape, expected value, and standard error) for samples of $n=36$ selected from a population with a mean of $\mu=100$ and a standard deviation of $\sigma=$ 12.

Ariana Nash
Ariana Nash
Numerade Educator
01:21

Problem 3

A sample is selected from a population with a mean of $\mu=80$ and a standard deviation of $\sigma=20$.
a. What is the expected value of $M$ and the standard error of $M$ for a sample of $n=4$ scores?
b. What is the expected value of $\mathrm{M}$ and the standard error of $\mathrm{M}$ for a sample of $n=16$ scores?

Sanchit Jain
Sanchit Jain
Numerade Educator
02:12

Problem 4

The distribution of sample means is not always a normal distribution. Under what circumstances will the distribution of sample means not be normal?

Diane Koenig
Diane Koenig
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01:21

Problem 5

A population has a standard deviation of $\sigma=30$.
a. On average, how much difference should exist between the population mean and the sample mean for $n=4$ scores randomly selected from the population?
b. On average, how much difference should exist for a sample of $n=25$ scores?
c. On average, how much difference should exist for a sample of $n=100$ scores?

Sanchit Jain
Sanchit Jain
Numerade Educator
01:21

Problem 6

For a population with a mean of $\mu=50$ and a standard deviation of $\sigma=10$, how much error, on average, would you expect between the sample mean $(M)$ and the population mean for:
a. a sample of $n=4$ scores
b. a sample of $n=16$ scores
c. a sample of $n=25$ scores

Sanchit Jain
Sanchit Jain
Numerade Educator
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Problem 7

For a population with a standard deviation of $\sigma=10$, how large a sample is necessary to have a standard error that is:
a. less than or equal to 5 points?
b. less than or equal to 2 points?
c. less than or equal to 1 point?

Keondre Parker
Keondre Parker
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Problem 8

If the population standard deviation is $\sigma=8$, how large a sample is necessary to have a standard error that is
a. less than 4 points?
b. less than 2 points?
c. less than 1 point?

Keondre Parker
Keondre Parker
Numerade Educator
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Problem 9

For a sample of $n=16$ scores, what is the value of the population standard deviation $(\sigma)$ necessary to have a standard error of
a. $\sigma_M=10$ points?
b. $\sigma_M=5$ points?
c. $\sigma_M=2$ points?

Keondre Parker
Keondre Parker
Numerade Educator

Problem 10

For a population with a mean of $\mu=60$ and a standard deviation of $\sigma=24$, find the $z$-score corresponding to each of the following samples.
a. $M=63$ for a sample of $n=16$ scores
b. $M=63$ for a sample of $n=36$ scores
c. $M=63$ for a sample of $n=64$ scores

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06:10

Problem 11

A sample of $n=25$ scores has a mean of $M=84$. Find the $z$-score for this sample:
a. If it was obtained from a population with $\mu=80$ and $\sigma=10$.
b. If it was obtained from a population with $\mu=80$ and $\sigma=20$.
c.If it was obtained from a population with $\mu=80$ and $\sigma=40$.

Ramon Kryzhan
Ramon Kryzhan
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Problem 12

A population forms a normal distribution with a mean of $\mu=80$ and a standard deviation of $\sigma=15$. For each of the following samples, compute the $z$-score for the sample mean and determine whether the sample mean is a typical, representative value or an extreme value for a sample of this size.
a. $M=84$ for $n=9$ scores
b. $M=84$ for $n=100$ scores

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01:21

Problem 13

A random sample is obtained from a normal population with a mean of $\mu=30$ and a standard deviation of $\sigma=8$. The sample
mean is $M=33$.
a. Is this a fairly typical sample mean or an extreme value for a sample of $n=4$ scores?
b. Is this a fairly typical sample mean or an extreme value for a sample of $n=64$ scores?

Sanchit Jain
Sanchit Jain
Numerade Educator
05:09

Problem 14

The population of IQ scores forms a normal distribution with a mean of $\mu=100$ and a standard deviation of $\sigma=15$. What is the probability of obtaining a sample mean greater than $M=105$,
a. for a random sample of $n=9$ people?
b. for a random sample of $n=36$ people?

Carly Stoner
Carly Stoner
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Problem 15

A population of scores forms a normal distribution with a mean of $\mu=75$ and a standard deviation of $\sigma=20$.
a. What proportion of the scores in the population have values less than $X=80$ ?
b. If samples of $n=4$ are selected from the population, what proportion of the samples will have means less than $M=80$ ?
c. If samples of $n=16$ are selected from the population, what proportion of the samples will have means less than $M=80$ ?

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Problem 16

A population of scores forms a normal distribution with a mean of $\mu=40$ and a standard deviation of $\sigma=12$.
a. What is the probability of randomly selecting a score less than $X=34$ ?
b. What is the probability of selecting a sample of $n=9$ scores with a mean less than $M=34$ ?
c. What is the probability of selecting a sample of $n=36$ scores with a mean less than $M=34$ ?

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Problem 17

A population of scores forms a normal distribution with a mean of $\mu=80$ and a standard deviation of $\sigma=10$.
a. What proportion of the scores have values between 75 and $85 ?$
b. For samples of $n=4$, what proportion of the samples will have means between 75 and 85 ?
c. For samples of $n=16$, what proportion of the samples will have means between 75 and 85 ?

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02:41

Problem 18

The population of SAT scores forms a normal distribution with a mean of $\mu=500$ and a standard deviation of $\sigma=100$. If the average SAT score is calculated for a sample of $n=25$ students,
a. What is the probability that the sample mean will be greater than $M=510$ ? In symbols, what is $p(M>510)$ ?
b. What is the probability that the sample mean will be greater than $M=520$ ? In symbols, what is $p(M>520)$ ?
c. What is the probability that the sample mean will be between $M=510$ and $M=520$ ? In symbols, what is $p(510<M<520)$ ?

Hailey Tomashek
Hailey Tomashek
Numerade Educator
04:20

Problem 19

The machinery at a food-packing plant is able to put exactly 12 ounces of juice in every bottle. However, some items such as apples come in variable sizes so it is almost impossible to get exactly 3 pounds of apples in a bag labeled " $3 \mathrm{lbs}$." Therefore, the machinery is set to put an average of $\mu=50$ ounces ( 3 pounds and 2 ounces) in each bag. The distribution of bag weights is approximately normal with a standard deviation of $\sigma=4$ ounces.
a. What is the probability of randomly picking a bag of apples that weighs less than 48 ounces ( 3 pounds)?
b. What is the probability of randomly picking $n=4$ bags of apples that have an average weight less than $M=48$ ounces?

Nick Johnson
Nick Johnson
Numerade Educator
03:21

Problem 20

The average age for licensed drivers in the county is $\mu=40.3$ years with a standard deviation of $\sigma=13.2$ years.
a. A researcher obtained a random sample of $n=16$ parking tickets and computed an average age of $M=38.9$ years for the drivers. Compute the $z$-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of $n=16$ people is a representative sample of licensed drivers?
b. The same researcher obtained a random sample of $n=36$ speeding tickets and computed an average age of $M=36.2$ years for the drivers. Compute the $z$-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of $n=36$ people is a representative sample of licensed drivers?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:49

Problem 21

People are selected to serve on juries by randomly picking names
from the list of registered voters. The average age for registered
voters in the county is μ = 44.3 years with a standard deviation of
σ = 12.4. A statistician computes the average age for a group of n
$=12$ people currently serving on a jury and obtains a mean of $M=$ 48.9 years.
a. How likely is it to obtain a random sample of $n=12$ jurors with an average age equal to or greater than 48.9 ?
b. Is it reasonable to conclude that this set of $n=12$ people is not a representative random sample of registered voters?

Victor Salazar
Victor Salazar
Numerade Educator
02:07

Problem 22

Welsh, Davis, Burke, and Williams (2002) conducted a study to evaluate the effectiveness of a carbohydrate-electrolyte drink on sports performance and endurance. Experienced athletes were given either a carbohydrate-electrolyte drink or a placebo while they were tested on a series of high-intensity exercises.
One measure was how much time it took for the athletes to run to fatigue. Data similar to the results obtained in the study are shown in the following table.
\begin{tabular}{lr}
Time to Run to Fatigue (in minutes) \\
& Mean $S E$ \\
Placebo & $21.7 \quad 2.2$ \\
Carbohydrate-electrolyte & $28.6 \quad 2.7$
\end{tabular}
a. Construct a bar graph that incorporates all of the information in the table.
b. Looking at your graph, do you think that the carbohydrateelectrolyte drink helps performance?

Hossam Mohamed
Hossam Mohamed
Numerade Educator