Practice Problems Notebook

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MAT241 Class

In this notebook, we’ll work through several practice problems. The problems appear in no particular order, and each one is best addressed using an inferential tool that we’ve encountered so far. As a reminder, we’ve seen

You may benefit by referring back to the General Strategy for Constructing Confidence Intervals, General Strategy for Conducting Hypothesis Tests, and Standard Error Decision Tree documents.

Note

This document contains “answers” in the form of approximate test statistics and p-values for hypothesis tests and bounds in the case of confidence intervals. Use the document cautiously since looking at the answers will tell you which tool is used to approach the scenario. Additionally, if you notice or suspect a typo, please post about it to the #general channel in our Slack workgroup.

Example Scenarios

Scenario 1: Average Commute Time

An urban planning committee wants to verify that the average commute time for citizens residing in a section of their city is higher than the national average of 30 minutes. They survey 40 residents, whose average commute time is 32 minutes with a standard deviation of 5 minutes. Test if the average commute time for city residents in this location is significantly greater than 30 minutes.

test statistic: ~2.53 p-value: ~0.0078

Scenario 2: Psychology Study on Sleep Quality

A psychology researcher wants to investigate whether a new sleep therapy program is effective in improving sleep quality. They collect data from 50 patients who completed the program and 50 who did not use the program. Among the program participants, 60% reported significantly improved sleep quality, while 48% of the control group reported improvement. Test for a significant difference in sleep improvement rates between the two groups.

test statistic: ~1.21 p-value: ~0.2263

Scenario 3: Effectiveness of Two Math Tutoring Methods

A school tests two methods for teaching math to see which is more effective. In Method A, 15 students achieve an average improvement of 12 points on a standardized test with a standard deviation of 4 points. In Method B, 18 students improve by an average of 14 points with a standard deviation of 3 points. Test if the two methods yield different average improvements at the \(\alpha = 0.10\) level of significance.

test statistic: ~-1.60 p-value: ~0.1319

Scenario 4: Marketing Campaign Success Rate

A company runs a social media campaign targeting two age groups: under 30 and over 30. Out of 150 users under 30, 45 clicked on the ad, while 40 out of 100 users over 30 clicked on it. Construct a 95% confidence interval for the difference in the click-through rates between the two age groups.

bounds: ~-0.0208 to 0.2208

Scenario 5: Free Throw Accuracy

A basketball coach tracks the free throw accuracy of two players. Both players shot 100 free throws under the same conditions. Player A made 85 while Player B made 77. Is there a statistically significant difference in free throw accuracy between the two players?

test statistic: ~1.45 p-value: ~0.1471

Scenario 6: Exercise Preferences

A fitness company wants to understand if there is an association between preferred exercise type and age group among their clients. They randomly survey 296 clients and categorize their preferences into three exercise types: Strength Training, Cardio, and Yoga/Pilates. The clients are also grouped by age: Under 30, 30-50, and Over 50. They summarize the results in the table below and ask you to determine whether an association between age group and exercise preference exists.

Age Group Strength Training Cardio Yoga/Pilates Total
Under 30 43 55 25 123
30-50 42 37 33 112
Over 50 21 22 18 61
Total 106 114 76 296

test statistic: ~4.6892 p-value: ~0.3207

Scenario 7: Sociology Survey on Pet Ownership

A survey of 300 randomly identified households in the US found that 200 households own pets. Construct a 90% confidence interval for the proportion of households that own pets.

bounds: 0.6218 to 0.7116

Scenario 8: Customer Waiting Time in a New Restaurant

A new restaurant claims that customers wait an average of 10 minutes to receive their orders. A random sample of 25 customers has an average waiting time of 12 minutes with a standard deviation of 3 minutes. Conduct a hypothesis test to determine if the actual average wait time at this location exceeds the restaurant’s claim.

test statistic: ~3.33 p-value: ~0.0014

Scenario 9: Recidivism Rates

In a study comparing recidivism rates for two rehabilitation programs, Program A has a recidivism rate of 22% (n = 120), while Program B has a rate of 18% (n = 100). Test for a difference in recidivism rates between the two programs at the 5% significance level.

test statistic: ~0.7661 p-value: ~0.4511

Scenario 10: Distribution of Car Colors

A car dealership wants to know if their sales align with national color preferences. Nationally, 30% of cars are white, 25% are black, 20% are gray, 15% are blue, and 10% are other colors. A sample of 500 cars sold at the dealership shows 150 white, 130 black, 110 gray, 80 blue, and 30 other cars. Test whether the dealership’s color distribution matches the national distribution.

test statistic: ~9.533 p-value: ~0.0491

Scenario 11: Plant Growth

A botanist tests a new fertilizer’s effect on plant growth. A sample of 25 plants grown with the fertilizer has an average height of 15 cm with a standard deviation of 3 cm. Construct a 95% confidence interval for the average height of plants grown with the fertilizer.

bounds: ~13.76 to ~16.24

Scenario 12: Blood Pressure Treatment Effectiveness

A study evaluates the effectiveness of two drugs in lowering blood pressure. Drug A shows an average reduction of 10 mmHg (n = 35, SD = 4 mmHg), while Drug B shows an average reduction of 8 mmHg (n = 40, SD = 5 mmHg). Construct a 99% confidence interval for the difference in blood pressure reduction between the two drugs.

bounds: ~-0.84 to ~4.84

Scenario 13: Housing Market Survey

In a city, 60% of a sample of 200 renters express interest in purchasing a home in the next year. Construct a 95% confidence interval for the proportion of renters interested in home ownership.

bounds: ~0.5321 to ~0.6679

Scenario 14: Ice Cream Flavor Preferences at a Festival

A food festival organizer wants to know if their customers’ preferences for ice cream flavors match the flavors ordered nationwide. Nationally, the distribution of favorite flavors is 35% vanilla, 25% chocolate, 20% strawberry, 10% mint chocolate chip, 4% cookie dough, and 6% other flavors. During the festival, a random sample of 750 ice cream orders shows 267 for vanilla, 196 for chocolate, 153 for strawberry, 71 for mint chocolate chip, 24 cookie dough, and 39 for other flavors. Conduct a chi-squared goodness of fit test to determine if the flavor preferences at the festival differ from the national distribution.

test statistic: ~2.74 p-value: ~0.74

Scenario 15: Vaccination Rates

A city health department wants to estimate the vaccination rate for a specific disease. In a sample of 400 residents, 312 have been vaccinated. Construct a 90% confidence interval for the vaccination rate.

bounds: ~0.7458 to ~0.8142

Scenario 16: Porch Pirating

Porch pirating is a term used to describe an unattended package being stolen from a doorstep, porch, etc. An analyst compares rates of porch pirating in two neighborhoods. In Neighborhood A, 24 of 100 households report at least one porch pirating incident within the last 12 months, while in Neighborhood B, 16 of 80 households reported these events over the same time period. Test for a difference in burglary rates between the two neighborhoods.

test statistic: ~0.6468 p-value: ~0.5178

Scenario 17: Athlete Recovery Time

Sports scientists measure the average heart rate recovery time (in seconds) for a group of 23 NCAA Division II athletes after a high-intensity workout. The average time to return to an individual’s normal heart rate after the workout is recovery time is 117 seconds with a standard deviation of 9 seconds. Test if the recovery time is significantly less than two minutes.

test statistic: ~-1.60 p-value: ~0.0619

Scenario 18: Running Speed Comparison

A coach tests the average running speed of two groups of athletes. Group A has an average speed of 7.2 m/s (n = 20, SD = 0.5), and Group B has an average speed of 7.0 m/s (n = 25, SD = 0.6). Construct a 95% confidence interval for the difference in average speed between the two groups.

bounds: ~-0.1433 to ~0.5433

Scenario 19: Study Habit Survey

A school surveys 500 students to see if study habits are associated with grade level (freshman, sophomore, junior, senior). Test for independence between grade level and whether students study regularly using a chi-squared test.

Grade Level Regular Studying
(ie. Weekly)		 Cramming or
No Studying		 Total
Freshman 70 55 125
Sophomore 80 45 125
Junior 95 30 125
Senior 70 55 125
Total 315 185 500

test statistic: ~14.37 p-value: 0.0024

Scenario 20: Memory Retention

A psychologist investigates the effect of caffeine on memory retention. In a sample of 30 students who consumed caffeine, the average recall score was 70% (SD = 8%). Construct a 99% confidence interval for the mean recall score for students who consumed caffeine.

bounds: ~0.4841 to ~0.9159

Scenario 21: Customer Satisfaction Survey

A store conducts a satisfaction survey with 400 customers, finding that 85% are satisfied. Construct a 95% confidence interval for the proportion of satisfied customers.

bounds: ~0.815 to ~0.885

Scenario 22: Online Course Completion Rates

A college compares completion rates between students enrolled in online and in-person classes. Out of 153 online students, 114 completed the course, while 147 out of 161 in-person students completed it. Test whether course completion rates for in-person students exceeds that for online students.

test statistic: 4.03 p-value: ~0.000028

Scenario 23: Streaming Service Usage

A media company surveys 350 randomly identified households from its member population and finds that 48% of them subscribe to a music streaming service. Construct a 90% confidence interval for the proportion of households that subscribe to music streaming.

bounds: ~0.4359 to ~0.5241

Scenario 24: College Students’ Weekly Study Hours

A university administrator wants to see if students at their institution study more than the national average of 15.5 hours per week. A random sample of 19 students from their school has an average weekly study time of 16.25 hours with a standard deviation of 2.3 hours. Test if the university’s students study more on average than students nationally.

test statistic: ~1.42 p-value: 0.0863

Scenario 25: Comparing Productivity in Two Office Settings

A company measures productivity (number of tasks completed) in two different office layouts, Layout A and Layout B. A sample of 20 employees working in Layout A completes an average of 30 tasks with a standard deviation of 6 tasks, while a sample of 25 employees in Layout B completes an average of 27 tasks with a standard deviation of 5 tasks. Use the \(\alpha = 0.10\) level of significance to test if there is evidence of a difference in productivity between the two office layouts.

test statistic: ~1.79 p-value: ~0.0636

Scenario 26: Customer Loyalty Program Impact

A company tests a new loyalty program. In a sample of 211 program participants, 124 made repeat purchases, while in a control group of 196 non-participants, 97 made repeat purchases. Conduct a test to determine whether the loyalty program is successful in generating repeat purchases.

test statistic: ~1.88 p-value: ~0.0301

Scenario 27: Traffic Violations by Age Group

A traffic study looks at the frequency of violations among two age groups. Among 18-25 year-olds, 43 of 155 drivers have received a violation in the past year, compared to 32 out of 127 drivers aged 26-35. Test for a difference in violation rates between the two age groups.

test statistic: ~0.4829 p-value: ~0.6292

Scenario 28: Exercise and Blood Pressure

A health study evaluates blood pressure reductions from exercise programs. The average blood pressure reduction for 25 participants in Program A was 12 mmHg (SD = 3 mmHg), and for 30 participants in Program B, it was 10 mmHg (SD = 4 mmHg). Construct a 95% confidence interval for the difference in blood pressure reduction between the two programs.

bounds: ~0.0493 to ~3.951