Confidence intervals and sample size | Practice Problems, Examples & Solutions

Confidence Intervals for the Mean When Is Known

Elementary Statistics

The methods of this section assume that sampling is from a population that is very large or infinite, and that we are sampling with replacement. If we have a relatively small population and sample withour replacement, we should modify $E$ to include a finite population correction factor, so that the margin of error is as shown in Exercise $37,$ where $N$ is the population size. That expression for the margin of error can be solved for $n$ to yield
$$
n=\frac{N \sigma^{2}\left(z_{\alpha / 2}\right)^{2}}{(N-1) E^{2}+\sigma^{2}\left(z_{\alpha / 2}\right)^{2}}
$$
Repeat Exercise $32,$ assuming that a simple random sample is selected without replacement from a population of 500 people. Does the additional information about the population size have much of an effect on the sample size?

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Known

01:08

Elementary Statistics

Find the indicated sample size.
You want to estimate the mean amount of annual tuition being paid by current full-time college students in the United States. First use the range rule of thumb to make a rough estimate of the standard deviation of the amounts spent. It is reasonable to assume that tuition amounts range from $\$ 0$ to about $\$40,000$. Then use that estimated standard deviation to determine the sample size corresponding to $95 \%$ confidence and a $\$ 100$ margin of error.

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Known

01:13

Elementary Statistics

Find the indicated sample size.
The Wechsler IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of scientists currently employed
by NASA. We want to be $95 \%$ confident that our ample mean is within five IQ points of the true mean. The mean for this population is clearly greater than $100 .$ The standard deviation for this population is probably less than 15 because it is a group with less variation than a group randomly selected from the general population; therefore, if we use $\sigma=15,$ we are being conservative by using a value that will make the sample size at least as large as necessary. Assume then that $\sigma=15$ and determine the required sample size.

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Known

Confidence Intervals for the Mean When Is Unknown

13 Practice Problems

04:37

Elementary Statistics

If a simple random sample of sixe $n$ is sclected without replacement from a finite population of sine $N,$ and the sample sixe is more than $5 \%$ of the population $\sin x(n \geq 0.05 N),$ better results an be obained by using the finite popubtion correction factor, which involves multiplying the margin of error $E$ by $\sqrt{(N-n) /(N-1)}$ For the smple of 100 weights of M\&M candics in Data Set 18 from Appendix B, we get
$\bar{x}=0.8565 \mathrm{g}$ and $s=0.0518 \mathrm{g}$. First construct a $95 \%$ confidence interval estimate of $\mu$ assuming that the population is large, then construct a $95 \%$ confidence interval cotimate of the mean weight of M\&CMs in the full bag from which the sample was taken. The full bag has $465 \mathrm{M} 8 \mathrm{kMs} .$ Compare the results.

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Not Known

03:39

Elementary Statistics

Appendix B Data Sets. Use the data sets from Appendix B.
Pulse Rates $A$ physician wants to develop criteria for determining whether a paticnt's pulxe rate is atypical, and she wants to determine whether there are significant differences between males and females. Use the sample pulse rates in Data Set 1 from Appendix $\mathbf{B}$.
a. Construct a $95 \%$ confidence interval estimate of the mean pulse rate for males.
b. Construct a $95 \%$ confidence interval estimate of the mean pulse rate for females.
c. Compare the preceding results. Can we conclude that the population means for males and females are different? Why or why not?

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Not Known

03:49

Elementary Statistics

Constructing Confidence Intervals. Construct the confidence interual.
Movie Lengths Listed below are 12 lengths (in minutes) of randomly sclected movics from Data Set 9 in Appendix B.
a. Construct a $99 \%$ confidence interval estimate of the mean length of all movics.
b. Assuming that it take 30 min to cmpty a thater affer a movic, clean it, allow time for the next audichce to chicr, and show previcus, what is the minimum time that a theater manager should plan berween start times of movics, assuming that this time will be sufficient for vypical movies?

Estimates and Sample Sizes

Estimating a Population Mean: $\sigma$ Not Known

Confidence Intervals and Sample size for Proportions

42 Practice Problems

0:00

Mathematical Statistics with Applications

Refer to Exercise 8.28. If the researcher wants to estimate the difference in proportions to within .05 with $90 \%$ confidence, how many graduates and nongraduates must be interviewed? (Assume that an equal number will be interviewed from each group.)

Estimation

Selecting the Sample Size

0:00

Mathematical Statistics with Applications

Let $Y$ be a binomial random variable with parameter $p$. Find the sample size necessary to estimate $p$ to within .05 with probability. 95 in the following situations:
a. If $p$ is thought to be approximately. 9
b. If no information about $p$ is known (use $p=.5$ in estimating the variance of $\hat{p}$ ).

Estimation

Selecting the Sample Size

06:09

Elementary Statistics

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9 - 1 along with "Table" answers based on Table $A$ - 3 with df equal to the smaller of $\boldsymbol{n}_{I}-\boldsymbol{I}$ and $\boldsymbol{n}_{2}-\boldsymbol{I} .$ )
Are Male Professors and Female Professors Rated Differently?
a. Use a 0.05 significance level to test the claim that two samples of course evaluation scores are from populations with the same mean. Use these summary statistics: Female professors:
$n=40, \bar{x}=3.79, s=0.51 ;$ male professors: $n=53, \bar{x}=4.01, s=0.53 .$ (Using the raw data in Data Set 17 "Course Evaluations" will yield different results.)
b. Using the summary statistics given in part (a), construct a $95 \%$ confidence interval estimate of the difference between the mean course evaluation score for female professors and male professors.
c. Example 1 used similar sample data with samples of size 12 and $15,$ and Example 1 led to the conclusion that there is not sufficient evidence to warrant rejection of the null hypothesis. Do the larger samples in this exercise affect the results much?

Inferences from Two Samples

Two Means: Independent Samples

Confidence Intervals for Variances and Standard Deviations

16 Practice Problems

0:00

Mathematical Statistics with Applications

In Exercise 8.97 . you derived upper and lower confidence bounds, each with confidence coefficient $1-\alpha,$ for $\sigma^{2} .$ How would you construct a $100(1-\alpha) \%$
a. upper confidence bound for $\sigma ?$
b. lower confidence bound for $\sigma$ ?

Estimation

Confidence Intervals for $\sigma^{2}$

0:00

Mathematical Statistics with Applications

The EPA has set a maximum noise level for heavy trucks at 83 decibels (dB). The manner in which this limit is applied will greatly affect the trucking industry and the public. One way to apply the limit is to require all trucks to conform to the noise limit. A second but less satisfactory method is to require the truck fleet's mean noise level to be less than the limit. If the latter rule is adopted, variation in the noise level from truck to truck becomes important because a large value of $\sigma^{2}$ would imply that many trucks exceed the limit, even if the mean fleet level were 83 dB. A random sample of six heavy trucks produced the following noise levels (in decibels):
$\begin{array}{llllll}85.4 & 86.8 & 86.1 & 85.3 & 84.8 & 86.0 .\end{array}$
Use these data to construct a $90 \%$ confidence interval for $\sigma^{2}$, the variance of the truck noiseemission readings. Interpret your results.

Estimation

Confidence Intervals for $\sigma^{2}$

03:05

Elementary Statistics

Using the heights and weights described in Exercise 1 , a height of $180 \mathrm{cm}$ is used to find that the predicted weight is $91.3 \mathrm{kg},$ and the $95 \%$ prediction interval is (59.0 kg. 123.6 kg). Write a statement that interprets that prediction interval. What is the major advantage of using a prediction interval instead of simply using the predicted weight of $91.3 \mathrm{kg} ?$ Why is the terminology of prediction interval used instead of confidence interval?

Correlation and Regression

Prediction Intervals and Variation