For questions 6 - 10 use the chi-squared distribution to test the hypothesis.
6) A restaurant owner wants to see if the business is good enough for him to purchase a restaurant. He asks the present owner for a breakdown of how many customers that come in for lunch each day and the results are as follows: Monday - 20, Tuesday - 30, Wednesday - 25, Thursday - 40 and Friday - 55. The prospective owner observes the restaurant and finds the following number of customers coming for lunch each day: Monday- 30, Tuesday - 15, Wednesday- 7, Thursday 40, and Friday - 33. At a 95% confidence level determine whether the present owner reported the correct number of customers for lunch each day.
7) An employer polled its employers to see if they agree with the proposed new store hours and whether or not their present shift made a difference in their answers. The customers answered 1 for agree, 2 for don't know, and 3 for disagree. Nine first shift employees answered "agree", 15 second shift employees answered "agree", and 20 third shift employees answered agree. With a 95% confidence level determine whether or not the employees' present shift played a role in their responses to the poll.
8) A politician surveyed 100 citizens to determine if their job title had anything to do with the way they responded to the following statement: "A city-wide curfew will be put into place. Select the time that you think it should be put into place. 8pm, 9pm, or 10pm". He is mostly concerned with the 10 pm responses. 25 teachers chose 10pm, 40 doctors chose 10pm, and 35 police responded 10pm. With a 95% confidence level, determine whether job title plays a role in how the citizens responded to the statement.
9) A meter reader did an experiment to see if there is a relationship between the number of tickets she writes and the number of blocks she is away from the park that is considered the heart of the city. At 0 blocks from the park she writes 35 tickets, at 1 block away from the park she writes 25 tickets, at 2 blocks from the park she writes 20 tickets and at 3 blocks from the park she writes 25 tickets. Use a 95% confidence level.
10) A high school principal asks his students to respond to the following statement: "School should start at 9:00am rather than 7:00am. Answer 1 for agree, 2 for don't know, and 3 for disagree." There were 90 seniors who answered agree, 35 juniors, 30 sophomores, and 25 freshmen. Help the principal decide with a 95% confidence level that the students' status played a role in how they responded to the question.
7) An employer polled its employers to see if they agree with the proposed new store hours and whether or not their present shift made a difference in their answers. The customers answered 1 for agree, 2 for don't know, and 3 for disagree. Nine first shift employees answered "agree", 15 second shift employees answered "agree", and 20 third shift employees answered agree. With a 95% confidence level determine whether or not the employees' present shift played a role in their responses to the poll.
DF: 3.841
X2: 20.54+20.54=41.08
41.08 > 3.841
We must reject the null hypothesis that the employees present shift played a role in the responses to the poll.
8) A politician surveyed 100 citizens to determine if their job title had anything to do with the way they responded to the following statement: "A city-wide curfew will be put into place. Select the time that you think it should be put into place. 8pm, 9pm, or 10pm". He is mostly concerned with the 10 pm responses. 25 teachers chose 10pm, 40 doctors chose 10pm, and 35 police responded 10pm. With a 95% confidence level, determine whether job title plays a role in how the citizens responded to the statement.
DF: 5.991
X2: 3.5034
5.991 > 3.5034
We can not reject the null hypothesis that job title plays a role in how the citizens responded to the statement about curfew change.
9) A meter reader did an experiment to see if there is a relationship between the number of tickets she writes and the number of blocks she is away from the park that is considered the heart of the city. At 0 blocks from the park she writes 35 tickets, at 1 block away from the park she writes 25 tickets, at 2 blocks from the park she writes 20 tickets and at 3 blocks from the park she writes 25 tickets. Use a 95% confidence level.
(35-26.25)²26.25+ (25-26.25)²26.25+(20-26.25)²26.25+(25-26.25)²26.25= 2.92+ .059+ 1.49+.059=4.52
95% confidence level = .05 with 3 degrees of freedom = 7.82
X² = 4.52 and 4.52 < 7.82
We must fail to reject the null hypothesis and conclude that there is no relationship between the number of tickets she writes and the number of blocks she is away from the park that is considered the heart of the city.
10) A high school principal asks his students to respond to the following statement: "School should start at 9:00am rather than 7:00am. Answer 1 for agree, 2 for don't know, and 3 for disagree." There were 90 seniors who answered agree, 35 juniors, 30 sophomores, and 25 freshmen. Help the principal decide with a 95% confidence level that the students' status played a role in how they responded to the question.
DF: 7.82
X2: 28.92
28.92 > 7.82
We must reject the null hypothesis that the students' status played a role in how they responded to the question.
To test the hypotheses in questions 6 - 10 using the chi-squared distribution, we need to perform goodness-of-fit tests or tests for independence, depending on the nature of the data. Let's go through each question and explain how to approach the test.
6) For question 6, we need to determine if the observed customer counts differ significantly from the reported counts. This requires a goodness-of-fit test. To perform this test, we follow these steps:
- Formulate the null hypothesis (H0): The reported counts of customers for each day are accurate.
- Define the alternative hypothesis (Ha): The reported counts of customers for each day are not accurate.
- Calculate the expected counts for each day by assuming the reported counts are accurate. This can be done by dividing the total observed customer count by 5 (for the 5 weekdays).
- Use the chi-square test statistic formula to compute the test statistic.
- Compare the test statistic to the critical value for the 95% significance level and determine if we reject or fail to reject the null hypothesis.
7) For question 7, we are interested in determining if the present shift of employees played a role in their responses. This requires a test for independence. Here are the steps:
- Formulate the null hypothesis (H0): There is no association between the employees' present shift and their responses.
- Define the alternative hypothesis (Ha): There is an association between the employees' present shift and their responses.
- Create a contingency table, with rows representing the shifts and columns representing the responses. Fill in the observed counts.
- Use the chi-square test statistic formula to compute the test statistic.
- Compare the test statistic to the critical value for the 95% significance level and determine if we reject or fail to reject the null hypothesis.
8) For question 8, we want to determine if job title plays a role in how citizens responded to the statement about the curfew. This also requires a test for independence. Here's how to approach it:
- Formulate the null hypothesis (H0): There is no association between job title and the citizens' responses to the curfew statement.
- Define the alternative hypothesis (Ha): There is an association between job title and the citizens' responses to the curfew statement.
- Create a contingency table with job titles as rows and curfew response options (in this case, focusing on the "10pm" response) as columns. Fill in the observed counts.
- Compute the test statistic using the chi-square test formula.
- Compare the test statistic to the critical value for the 95% significance level and determine if we reject or fail to reject the null hypothesis.
9) For question 9, we want to investigate if there is a relationship between the number of tickets written and the distance from the park. We'll use a goodness-of-fit test. Follow these steps:
- Formulate the null hypothesis (H0): There is no relationship between the number of tickets written and the distance from the park.
- Define the alternative hypothesis (Ha): There is a relationship between the number of tickets written and the distance from the park.
- Calculate the total number of tickets and determine the expected proportions based on the assumption of no relationship.
- Set up a contingency table with the observed and expected counts.
- Calculate the test statistic using the chi-square test formula.
- Compare the test statistic to the critical value for the 95% significance level and determine if we reject or fail to reject the null hypothesis.
10) For question 10, we want to determine if the students' status (senior, junior, sophomore, or freshman) played a role in how they responded to the statement. This requires a test for independence. Here's the step-by-step process:
- Formulate the null hypothesis (H0): There is no association between students' status and their responses.
- Define the alternative hypothesis (Ha): There is an association between students' status and their responses.
- Create a contingency table with students' status as rows and the response options as columns. Fill in the observed counts.
- Calculate the test statistic using the chi-square test formula.
- Compare the test statistic to the critical value for the 95% significance level and determine if we reject or fail to reject the null hypothesis.
Remember, the specific calculations and critical values will depend on the sample sizes and degrees of freedom for each test. Make sure to consult a chi-square distribution table or use statistical software to find the appropriate values.
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See response to your later post.