For questions 1- 5 use confidence intervals to test the hypothesis.
1) A light bulb producing company states that its lights will last an average of 1200 hours with a standard deviation of 200 hours. A sample of 100 light bulbs from the company were tested and the researcher found that the average life of each light bulb was 1050 hours. At a 95% confidence level, determine whether these light bulbs are in compliance with the company's claim.
2) A company's human resource department claims that all employees are present on the average 4 days out of the work week with a standard deviation of 1. They hired an outside company to do an audit of their employees' absences. The company took a sample a 10 people and found that on the average the employees were present 3 days per week. With a 95% confidence level, determine whether the company's claim is true based on the data from the sample.
3) A teacher claims that all of her students pass the state mandated test with an average of 90 with a standard deviation of 10. The principal gave the test to 20 of her students to see if the teacher's claim was true. He found that the average score was 75. With a 95% confidence level, determine whether the teacher is making the correct claim about all of her students.
4) The lifeguard's at a local pool have to be able to respond to a distressed swimmer at an average of 10 seconds with a standard deviation of 4 in order to be considered for employment. If a sample of 100 lifeguards showed that their average response time is 15 seconds, with a confidence level of 95% determine whether this group may be considered for employment.
5) It is believed that an average of 20 mg of iodine is in each antibiotic cream produced by a certain company with a standard deviation of 5 mg. The company pulled 150 of its antibiotic creams and found that on the average each cream contained 29 mg of iodine. Determine with a 95% confidence level whether or not these creams are in compliance with the company's belief?
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.
To test the hypothesis using confidence intervals, you can follow these steps:
1) Calculate the sample mean: For each question, you are given the sample mean.
2) Calculate the standard error: The standard error is the standard deviation divided by the square root of the sample size. In these questions, the standard deviation is given. So, divide the standard deviation by the square root of the sample size to find the standard error.
3) Determine the margin of error: The margin of error is calculated by multiplying the critical value (obtained from the confidence level) by the standard error. The critical value corresponds to the desired confidence level. For example, a 95% confidence level corresponds to a critical value of 1.96 (for a normal distribution).
4) Calculate the confidence interval: The confidence interval is calculated by subtracting the margin of error from the sample mean and adding it to the sample mean. This will give you the lower and upper bounds of the confidence interval.
5) Analyze the results: If the hypothesized value (from the claim or statement) falls within the confidence interval, then we fail to reject the hypothesis. If the hypothesized value falls outside the confidence interval, then we reject the hypothesis.
Now let's apply these steps to each question:
1) Sample mean = 1050, standard deviation = 200, sample size = 100, confidence level = 95%
- Calculate standard error: standard error = standard deviation / sqrt(sample size)
- Calculate margin of error: margin of error = critical value * standard error (critical value = 1.96 for 95% confidence level)
- Calculate the confidence interval: lower bound = sample mean - margin of error, upper bound = sample mean + margin of error
- Analyze the results: If the range 1200 (company's claim) falls within the confidence interval, the light bulbs are in compliance with the company's claim.
Repeat the above steps for questions 2-5 using the given information in each question.
For questions 6-10, you will be using the chi-squared distribution. To test the hypothesis using the chi-squared distribution, follow these steps:
1) Set up the null and alternative hypotheses: The null hypothesis assumes that there is no difference or relationship, while the alternative hypothesis assumes that there is a difference or relationship. The null hypothesis is usually denoted as H0, and the alternative hypothesis is denoted as Ha.
2) Calculate the test statistic: The test statistic for the chi-squared distribution is calculated based on the observed frequencies and the expected frequencies. The expected frequencies can be calculated based on the null hypothesis.
3) Determine the critical value: The critical value corresponds to the desired significance level and the degrees of freedom. Degrees of freedom depend on the number of categories or groups being compared.
4) Compare the test statistic with the critical value: If the test statistic is greater than the critical value, then reject the null hypothesis. If the test statistic is less than the critical value, then fail to reject the null hypothesis.
Now let's apply these steps to each question:
6) Set up the null and alternative hypotheses: The null hypothesis assumes that the present owner reported the correct number of customers for lunch each day, while the alternative hypothesis assumes there is a difference.
- Calculate the test statistic: This involves calculating the chi-squared value based on the observed and expected frequencies.
- Determine the critical value: Chi-squared critical value can be obtained from the chi-squared distribution table based on the desired confidence level and degrees of freedom.
- Compare the test statistic with the critical value: If the test statistic is greater than the critical value, reject the null hypothesis.
Repeat the above steps for questions 7-10 using the given information in each question.