Testing the difference between two means, two proportions, and two variances | Practice Problems, Examples & Solutions

Testing the Difference Between Two Means: Using the z Test

Elementary Statistics a Step by Step Approach

Commuting Times for College Students The mean travel time to work for Americans is 25.3 minutes. An employment agency wanted to test the mean commuting times for college graduates and those with only some college. Thirty-five college graduates spent a mean time of 40.5 minutes commuting to work with a population variance of 67.24 . Thirty workers who had completed some college had a mean commuting time of 34.8 minutes with a population variance of $39.69 .$ At the 0.05 level of significance, can a difference in means be concluded?

Testing the Difference Between Two Means, Two Proportions, and Two Variances

Testing the Difference Between Two Means: Using the z Test

06:42

Elementary Statistics a Step by Step Approach

Commuting Times The U.S. Census Bureau reports that the average commuting time for citizens of both Baltimore, Maryland, and Miami, Florida, is approximately 29 minutes. To see if their commuting times appear to be any different in the winter, random samples of 40 drivers were surveyed in each city and the average commuting time for the month of January was calculated for both cities. The results are shown. At the 0.05 level of significance, can it be
concluded that the commuting times are different in the winter?
$$
\begin{array}{lcc}{} & {\text { Miami }} & {\text { Baltimore }} \\ \hline \text { Sample size } & {40} & {40} \\ {\text { Sample mean }} & {28.5 \min } & {35.2 \mathrm{min}} \\ {\text { Population standard deviation }} & {7.2 \mathrm{min}} & {9.1 \mathrm{min}}\end{array}
$$

Testing the Difference Between Two Means, Two Proportions, and Two Variances

Testing the Difference Between Two Means: Using the z Test

03:24

Elementary Statistics a Step by Step Approach

Explain the difference between testing a single mean and testing the difference between two means.

Testing the Difference Between Two Means, Two Proportions, and Two Variances

Testing the Difference Between Two Means: Using the z Test

Testing the Difference Between Two Means of Independent Samples: Using the t Test

29 Practice Problems

12:27

Statistics Informed Decisions Using Data

A physical therapist believes that women are more flexible than men. She measures the flexibility of 31 randomly selected women and 45 randomly selected men by determining the number of inches subjects could reach while sitting on the floor with their legs straight out and back perpendicular to the ground. The more flexible an individual is, the higher the measured flexibility will be. After entering the data into MINITAB, she obtained the following results:
(TABLE CAN'T COPY)
(a) State the null and alternative hypotheses.
(b) Identify the $P$ -value and state the researcher's conclusion if the level of significance was $\alpha=0.01$
(c) What is the $95 \%$ confidence interval for the mean difference in flexibility of men versus women? Interpret this interval.

Inferences on Two Samples

Inference about Two Means: Independent Samples

01:44

Statistics Informed Decisions Using Data

The median length of stay for male substance-abuse outpatient treatment completer is 105 days. The following data represent the length of stays for a random sample of female substance-abuse outpatient treatment completer.
Using the Wilcoxon signed-ranks test at the $\alpha=0.05$ level of significance, does the median length of stay seem different for males and females?
(TABLE CAN'T COPY)

Nonparametric Statistics

Inferences about the Difference between Two Medians: Independent Samples

01:34

Statistics Informed Decisions Using Data

A researcher wants to know if the octane level of gasoline affects the gas mileage of a
-car. She randomly selects 10 cars and puts 5 gallons of 87 -octane gasoline in the tank. On a closed track, each car is driven at 50 miles per hour until it runs out of gas. The experiment is repeated, with each car getting 5 gallons of 92 -octane gasoline.
- The miles per gallon for each car are then computed. The results are shown. Would you recommend purchasing 92 octane? Why or why not?
(TABLE CAN'T COPY)

Nonparametric Statistics

Inferences about the Difference between Two Medians: Independent Samples

Testing the Difference Between Two Means: Dependent Samples

12 Practice Problems

03:44

Statistics Informed Decisions Using Data

Hardness Testing The manufacturer of hardness testing equipment uses steel-ball indenters to penetrate metal that is being tested. However, the manufacturer thinks it would be better to use a diamond indenter so that all types of metal can be tested. Because of differences between the two types of indenters, it is suspected that the two methods will produce different hardness readings. The metal specimens to be tested are large enough so that two indentions can be made. Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings. Construct a $95 \%$ confidence interval to judge whether Two indepters result in different measurements.
(TABLE CANNOT COPY)

Inferences on Two Samples

Inference about Two Means: Dependent Samples

02:34

Statistics Informed Decisions Using Data

Caffeine-Enhanced Workout? since its removal from the banned substances list in 2004 by the World Anti-Doping Agency, caffeine has been used by athletes with the expectancy that it enhances their workout and performance. Many studies have been conducted to assess the effect of caffeine on athletes, but few look at the role it plays in sedentary females. Researchers at the University of Western Australia conducted a test in which they determined the rate of energy expenditure (kilojoules) on 10 healthy, sedentary females who were nonregular caffeine users. Each female was randomly assigned either a placebo or caffeine pill (6 $\mathrm{mg} / \mathrm{kg}$ ) 60 minutes prior to exercise. The subject rode an exercise bicycle for 15 minutes at $65 \%$ of their maximum heart rate, and the energy expenditure was measured. The process was repeated on a separate day for the remaining treatment. The mean difference in energy expenditure (caffeine-placebo) was 18 kJ with a standard deviation of 19 kJ.
(a) State the null and alternative hypothesis to determine if caffeine increases energy expenditure.
(b) Assuming the differences are normally distributed, determine if caffeine appears to increase energy expenditure at the $\alpha=0.05$ level of significance.

Inferences on Two Samples

Inference about Two Means: Dependent Samples

00:55

Statistics Informed Decisions Using Data

A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched-pairs data. If the observations from sample 1 are $X_{i}$ and the observations from sample 2 are $Y_{i},$ and $d_{i}=X_{i}-Y_{i},$ then the null hypothesis is $H_{0}: \mu_{d}=0$ and the alternative hypothesis is $H_{1}: \mu_{d}$
-0

Inferences on Two Samples

Inference about Two Means: Dependent Samples

Testing the Difference Between Proportions

15 Practice Problems

04:18

Elementary Statistics

To test the null hypothesis that the difference be tween two population proportions is equal to a nonzero constant $c$, use the test statistic
$$z=\frac{\left(\hat{p}_{1}-\hat{p}_{2}\right)-c}{\sqrt{\frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}}}}$$
As long as $n_{1}$ and $n_{2}$ are both large, the sampling distribution of the test statistic $z$ will be approximately the standard normal distribution. Refer to Exercise 27 and use a 0.01 significance level to test the claim that the rate of thyroid disease among female atom bomb survivors is equal to 15 percentage points more than that for male atom bomb survivors.

Inference From Two Samples

Inferences About Two Proportions

03:15

Elementary Statistics

Use the sample data in Exercise 35 with a 0.05 significance level to test the claim that the percentage of women who say that female bosses are harshly critical is greater than the percentage of men. Does the significance level of 0.05 used in this exercise correspond to the $95 \%$ confidence level use for the preceding exercise? Considering the sampling method, is the hypothesis test valid?

Inference From Two Samples

Inferences About Two Proportions

02:41

Elementary Statistics

Tax returns include an option of designating $\$ 3$ for presidential election campaigns, and it does not cost the taxpayer anything to make that designation. In a simple random sample of 250 tax returns from $1976,27.6 \%$ of the returns designated the $\$ 3$ for the campaign. In a simple random sample of 300 recent tax returns, $7.3 \%$ of the returns designated the $\$ 3$ for the campaign (based on data from U S A Today. Use a 0.01 significance level to test the claim that the percentage of returns designating the $\$ 3$ for the campaign was greater in 1976 than it is now.

Inference From Two Samples

Inferences About Two Proportions

Testing the Difference Between Two Variances

9 Practice Problems

02:43

Elementary Statistics

Discrimination The Revenue Commissioners in Irdand conducted a contest for promotion. Ages of the unsuccessful and successful applicants are given below (based on data from "Debating the Use of Statistical Evidence in Allegations of Age Discrimination," by Barry and Boland, American Statisticion, Vol. $58,$ No. 2 ). Use a 0.05 significance level to test the daim that both samples are from populations having the same standard deviation.
$$\begin{array}{lllllllllllll}
\text { Unsuccessful applicants } & 34 & 37 & 37 & 38 & 41 & 42 & 43 & 44 & 44 & 45 & 45 & 45 \\
& 46 & 48 & 49 & 53 & 53 & 54 & 54 & 55 & 56 & 57 & 60 &
\end{array}$$
$$\begin{array}{lllllllllllll}
\text { Successful applicants } & 27 & 33 & 36 & 37 & 38 & 38 & 39 & 42 & 42 & 43 & 43 & 44 \\
& 44 & 44 & 45 & 45 & 45 & 45 & 46 & 46 & 47 & 47 & 48 & 48
\end{array}$$

Inference From Two Samples

Comparing Variation in Two Samples

02:29

Elementary Statistics

Using the ample data from Data Set 23 in Appendix B. 21 homes with living areas under $2000 \mathrm{ft}^{2}$ have selling prices with a standard deviation of $\$ 32,159.73.$ There are 19 homes with living areas greater than $2000 \mathrm{ft}^{2}$ and they have selling prices with a standard deviation of $\$ 66,628.50 .$ Use a 0.05 significance level to test the claim of a real cotate agent that homes larger than $2000 \mathrm{ft}^{2}$ have selling prices that vary more than the smaller homes.

Inference From Two Samples

Comparing Variation in Two Samples

02:14

Elementary Statistics

Hypothesis Tests of Qaims About Variation. Test the given claim. Assume that both samples are independent simple random samples from populations having normal distributions.
9. Baseline Characteristics In journal articles about dinical experiments, it is common to include bascline chanarcristic of the different treatment groups so that they an be compared. In an artide about the cffects of different dicts, a table of bascline characteristics showed that
40 subjects treated with the Atkins diet had a mean age of 47 years with a standard deviation
of 12 years. Also, 40 subjects trated with the Zone dict had a mean age of 51 years with a standard deviation of 9 years. Use a 0.05 significance level to test the daim that subjects from both tratment groups have ages with the same amount of variation. How are comparisons of treatments affected if the treatment groups have different characteristics?

Inference From Two Samples

Comparing Variation in Two Samples