AnswersCA-Statistics and Probability BIntroduction to Hypothesis Testing

Introduction to Hypothesis Testing — Unit test Answers

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A carnival game is designed so that approximately 10% of players will win a large prize. If there is evidence that the percentage differs significantly from this target, then adjustments will be made to the game. To investigate, a random sample of 100 players is selected from the large population of all players. Of these players, 19 win a large prize. The question of interest is whether the data provide convincing evidence that the true proportion of players who win this game differs from 0.10. Are the conditions for inference met for conducting a z-test for one proportion?

A
Yes, the random, 10%, and large counts conditions are all met.
B
No, the random condition is not met.
C
No, the 10% condition is not met.
D
No, the large counts condition is not met.
2
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A plant in Alamo, TN, manufactures complex transformer components that must meet specific guidelines for safety. One such component is constructed to deliver 1,000 volts of electricity. A component creates a critical safety hazard if it absorbs humidity at a level above 3%. Any components that absorb too much humidity will be destroyed. A quality control inspector uses a random sample of components to conduct a hypothesis test with H0: The humidity level absorbed is 3%, and Ha: The humidity level absorbed is more than 3%. What is the consequence of a Type II error in this context?

A
The company believes the humidity absorbed is more than 3% when in fact it is not. Correctly functioning components will be destroyed at great expense to the company.
B
The company believes the voltage delivered is more than 1,000 volts, when in fact it is not more than 1,000 volts. Correctly functioning components will be destroyed at great expense to the company.
C
The company believes the voltage delivered is no more than 1,000 volts, when in fact it is more than 1,000 volts. The company will sell components that absorb a dangerous level of humidity.
D
The company believes the humidity absorbed is not more than 3%, when in fact it is more than 3%. The company will sell components that absorb a dangerous level of humidity.
3

A school guidance counselor is concerned that a greater proportion of high school students are working part-time jobs during the school year than a decade ago. A decade ago, 28% of high school students worked a part-time job during the school year. To investigate whether the proportion is greater today, a random sample of 80 high school students is selected. It is discovered that 37.5% of them work part-time jobs during the school year. The guidance counselor would like to know if the data provide convincing evidence that the true proportion of all high school students who work a part-time job during the school year is greater than 0.28. What are the appropriate hypotheses for this test?

A
H0: p = 0.28 versus Ha: p < 0.28, where p = the proportion of all high school students who work a part-time job during the school year.
B
H0: p = 0.28 versus Ha: p > 0.28, where p = the proportion of all high school students who work a part-time job during the school year.
C
H0: p = 0.28 versus Ha: p < 0.28, where p = the proportion of high school students in the sample who work a part-time job during the school year.
D
H0: p = 0.28 versus Ha: p > 0.28, where p = the proportion of high school students in the sample who work a part-time job during the school year.
4

Simone read online that the failure rate in Arizona for the first attempt of the written driver’s test is 60%. Simone thinks the Arizona rate is less than 60%. To investigate, she selects an SRS of 50 Arizona drivers and finds that 27 failed their first written driving test. To determine if this provides convincing evidence that the failure rate for Arizona is less than 60%, 200 trials of a simulation are conducted. Simone’s hypotheses are: H0: p = 60% and Ha: p < 60%, where p = the true proportion of Arizona drivers who fail the first attempt of the written driver’s test. Based on the results of the simulation, the estimated P-value of this test is 0.035. Using a = 0.05, what conclusion should Simone reach?

A
Because the P-value of 0.035 < a, Simone should reject H0. There is convincing evidence that the Arizona written driver’s test has a true first-attempt failure rate less than 60%.
B
Because the P-value of 0.035 < a, Simone should reject H0. There is not convincing evidence that the Arizona written driver’s test has a true first-attempt failure rate less than 60%.
C
Because the P-value of 0.035 < a, Simone should fail to reject H0. There is convincing evidence that the Arizona written driver’s test has a true first-attempt failure rate less than 60%.
D
Because the P-value of 0.035 < a, Simone should fail to reject H0. There is not convincing evidence that the Arizona written driver’s test has a true first-attempt failure rate less than 60%.
5

One of Natalia’s professors claims that 13% of the students at their college frequently wear hats. Natalia feels that the actual rate is much lower. She selects a random sample of 100 students at the college and finds that only 9 of them wear hats frequently. To determine if her data provide convincing evidence that the true proportion is less than 13%, she conducts 200 trials of a simulation. Natalia is testing the hypotheses: H0: p = 13% and Ha: p < 13%, where p = the true proportion of students at the college who frequently wear hats. Based on the results of the simulation, what is an estimate of the P-value of the test?

Question illustration
A
0.005
B
0.01
C
0.077
D
0.09
6

It is common knowledge that a fair penny will land heads up 50% of the time and tails up 50% of the time. It is very unlikely for a penny to land on its edge when flipped, so a probability of 0 is assigned to this outcome. A curious student suspects that 5 pennies glued together will land on their edge 50% of the time. To investigate this claim, the student securely glues together 5 pennies and flips the penny stack 100 times. Of the 100 flips, the penny stack lands on its edge 46 times. The student would like to know if the data provide convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from 0.5. The student tests the hypotheses H0: p = 0.50 versus Ha: p ≠ 0.50, where p = the true proportion of all flips for which the penny stack will land on its edge. The conditions for inference are met. The standardized test statistic is z = –0.80 and the P-value is 0.2119. What conclusion should the student make using the α = 0.10 significance level?

A
Because the test statistic is less than α = 0.10, there is convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from 0.5.
B
Because the P-value is greater than α = 0.10, there is convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from 0.5.
C
Because the P-value is greater than α = 0.10, there is not convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from 0.5.
D
Because the test statistic is less than α = 0.10, there is not convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from 0.5.
7

A school guidance counselor is concerned that a greater proportion of high school students are working part-time jobs during the school year than a decade ago. A decade ago, 28% of high school students worked a part-time job during the school year. To investigate whether the proportion is greater today, a random sample of 80 high school students is selected. It is discovered that 37.5% of them work part-time jobs during the school year. The guidance counselor would like to know if the data provide convincing evidence that the true proportion of all high school students who work a part-time job during the school year is greater than 0.28.

A
If the true proportion of high school students who work a part-time job during the school year is p = 0.747, there is a 0.59 probability that the guidance counselor will find convincing evidence for Ha: p > 0.28.
B
If the true proportion of high school students who work a part-time job during the school year is p = 0.747, there is a 0.59 probability that the guidance counselor will find convincing evidence for Ha: p = 0.28.
C
If the true proportion of high school students who work a part-time job during the school year is p = 0.28, there is a 0.41 probability that the guidance counselor will find convincing evidence for Ha: p > 0.28.
D
If the true proportion of high school students who work a part-time job during the school year is p = 0.28, there is a 0.41 probability that the guidance counselor will find convincing evidence for Ha: p = 0.28.
8

Animal shelters in a county need at least 15% of their animals to be adopted weekly to have room for the new animals that are brought into the various shelters. The county manager takes a random sample of shelters each week to estimate the overall proportion of animals that are adopted. If he concludes that the proportion has dropped below 15%, he will not accept any new animals into the shelters that week. He tests the hypotheses: H0: The adoption rate is 15%, and Ha: The adoption rate is less than 15%. What is a Type I error, and what is its consequence in this context?

A
The manager believes the adoption rate is still 15%, when it actually has dropped below 15%. The manager will accept more animals into the shelters and will run out of room.
B
The manager believes the adoption rate has dropped below 15%, when it actually has not. The manager will accept more animals into the shelters and will run out of room.
C
The manager believes the adoption rate has dropped below 15%, when it actually has not. The manager will not accept more animals into the shelters, when there actually is room to care for those animals.
D
The manager believes the adoption rate is still 15%, when it actually has dropped below 15%. The manager will not accept some animals into the shelters, thinking there will not be enough room, when they could have taken care of those animals.
9

It is claimed that 75% of puppies are house-trained by the time they are 6 months old. To investigate this claim, a random sample of 50 puppies is selected. It is discovered that 42 are house-trained by the time they are 6 months old. A trainer would like to know if the data provide convincing evidence that greater than 75% of puppies are house-trained by the time they are 6 months old. Are the conditions for inference met?

A
Yes, the conditions for inference are met.
B
No, one condition is not met.
C
No, two conditions are not met.
D
No, three conditions are not met.
10

A political pollster claims that 55% of voters prefer candidate A. To investigate this claim, a random sample of 75 voters is polled. The pollster finds that 39 of those polled prefer candidate A. He would like to know if the data provide convincing evidence that the true proportion of all voters who prefer candidate A is less than 55%. What are the values of the test statistic and P-value for this test?Find the z-table here.

A
z = StartStartFraction 0.52 minus 0.55 OverOver StartRoot StartFraction 0.52 (1 minus 0.52) Over 75 EndFraction EndEndFraction, p-value = 0.3015
Option A
B
z = StartStartFraction 0.52 minus 0.55 OverOver StartRoot StartFraction 0.52 (1 minus 0.52) Over 75 EndFraction EndEndFraction, p-value = 0.6030
Option B
C
z = StartStartFraction 0.52 minus 0.55 OverOver StartRoot StartFraction 0.55 (1 minus 0.55) Over 75 EndFraction EndEndFraction, p-value = 0.3015
Option C
D
z = StartStartFraction 0.52 minus 0.55 OverOver StartRoot StartFraction 0.55 (1 minus 0.55) Over 75 EndFraction EndEndFraction, p-value = 0.6030
Option D
11

The owner of a popular coffee shop believes that customers who drink espresso are less likely to use their own cup compared to customers who drink coffee. Customers using their own cups get a 5% discount, which is displayed on the receipt. The owner randomly selects 50 receipts from all espresso purchases and 50 receipts from all coffee purchases. For espresso purchases, 15 receipts showed that the customer used their own cup. For coffee purchases, 24 receipts showed the customer used their own cup. Let pEspresso = the true proportion of customers who drink espresso and use their own cup and pCoffee = the true proportion of customers who drink coffee and use their own cup. Which of the following are the correct hypotheses to test the owner’s claim?

A
Option
Option A
B
Option
Option B
C
Option
Option C
D
Option
Option D
12

A claw machine game displays a sign that claims 90% of plays result in a win. A suspicious customer watches 40 consecutive plays of this game and observes that there is a winner in 30 of the games. The customer would like to know if the data provide convincing evidence that the true proportion of winners is less than 0.9. Are the conditions for inference met?

A
Yes, the conditions for inference are met.
B
No, one condition is not met.
C
No, two conditions are not met.
D
No, three conditions are not met.
13

A political pollster claims that 55% of voters prefer candidate A. To investigate this claim, a random sample of 75 voters is polled. The pollster finds that 39 of those polled prefer candidate A. He would like to know if the data provide convincing evidence that the true proportion of all voters who prefer candidate A is less than 55%. The P-value of this test is 0.3015. Interpret the P-value.

A
Assuming that the true population proportion of voters who prefer candidate A is 0.55, the probability of obtaining a sample statistic of 0.52 in a random sample of 75 voters is 0.3015.
B
Assuming that the true population proportion of voters who prefer candidate A is 0.55, the probability of obtaining a sample statistic of 0.52 or less in a random sample of 75 voters is 0.3015.
C
Assuming that the true population proportion of voters who prefer candidate A is less than 0.55, the probability of obtaining a sample statistic of 0.52 in a random sample of 75 voters is 0.3015.
D
Assuming that the true population proportion of voters who prefer candidate A is less than 0.55, the probability of obtaining a sample statistic of 0.52 or less in a random sample of 75 voters is 0.3015.
14

A local candidate believes more than half of the constituents in their district would favor them for political office. A random sample of 120 voters was polled, and 65 stated they would likely vote for the candidate. Which hypotheses would test the candidate’s claim?

A
H 0: p hat = 0.54, H alpha: p hat greater-than 0.54
Option A
B
H 0: p = 0.50, H alpha: p less-than 0.50
Option B
C
H 0: p = 0.54, H alpha: p greater-than 0.54
Option C
D
H 0: p = 0.50, H alpha: p greater-than 0.50
Option D
15

A teacher has a large container of blue, red, and green beads. She reports to the students that the proportion of blue beads in the container is 0.30. The students feel the proportion of blue beads is lower than 0.30. A student randomly selects 60 beads and finds that 12 of the beads are blue. The P-value for the test of the hypotheses, and , is 0.045. What is the correct conclusion given ?

Question illustration
A
Because the P-value is less than , the student should reject H0.
Option A
B
Because the P-value is less than , the student should fail to reject H0.
Option B
C
Because the P-value is greater than , the student should reject H0.
Option C
D
Because the P-value is greater than , the student should fail to reject H0.
Option D

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