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**Lab 4 | Complete Solution**

- From Mathematics, Statistics

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Lab 4

Objectives:

• Learn to use Minitab to get confidence interval estimates for population proportions.

• You will have to recall how you computed confidence intervals for proportions and means from earlier work.

• While Minitab will do the calculation for you and get the interval, you are expected to set the problem up carefully, use appropriate sampling distribution, justify that and interpret the confidence interval.

1. a. The file tvhrs.mtw includes the results of a study of TV-viewing patterns of a random group of 120 individuals. Open the file in Minitab. The variables in the study are;

Column Name Description C1 ID Subject Identification # C2 Age Group Age group of subject C3 Age Age C4 Gender Gender : Male or Female C5 Sesame Whether or not the subject watched Sesame Street C6 HrsTV Hours spent watching TV in a week C7 hrsMTV Hours spent watching MTV in a week C8 HrsNews Hours spent watching news in a week C9 Educ

Highest level of Education achieved

(a) Use Minitab to compute a 97% confidence interval for the average number of hours a randomTV viewer spends watching news in a week.

(b) Use Minitab to compute the proportion of TV viewers who watch Sesame Street. Note that

"success" in this case is an individual watching Sesame

Street, so a response of "yes" in the survey.

(c) Let us compare the average number of hours spent watching MTV between men and women. First we need to break up the data by gender. You can do this easily by using using the menu option: Data UnStack Columns. In the dialog box, enter the column (HrsMTV) that needs to be unstacked; this is to be broken up by the gender. The gender is called "subscripts". So, in the box for "Using subscripts in", enter the column that has the Gender information. Store the unstacked data "after the last column in use." and check the box to make sure that you "name the columns containing the unstacked data." You should now have two extra columns for the number of hours MTV is watched one for men and one for women. Now compute a 95% confidence interval for the average number of hours MTV is watched by a male and a female.

2. An article in J. of Rail and Rapid Transit presented the following data on time to repair (in minutes) a rail break in the high rail on a curved track of a certain railway line:

159, 120, 480,149, 270,547, 340, 43, 228, 202, 240, 218

Compute an appropriate 90% confidence interval for the true average time to repair a rail break on this high rail.

Hypotheses Testing

(a) Testing population proportion:

Suppose we want to test H0: p=0.3 against H1: p > 0.3, where p is the proportion of "successes" in the population. Suppose we use a sample of size 200. Suppose 75 in the sample of 200 were "successes". Suppose our level of significance (LOS) is 0.02.

Question 1: What is the parameter that you are testing here? What is the point estimate of the parameter, based on your sample. Use appropriate notation.

Question 2: What type of test is this? In which direction do you have the "rejection region."

Question 3: Compute and interpret the P-value; Write your conclusion regarding your hypotheses test. Make sure that you copy the hypotheses to your document and also follow the hypotheses testing process. Minitab Steps:

(i) Use Stat → Basic statistics → 1 Proportion,

(ii) click on Summarized data and enter the values for the number of trials (this is the sample size) and number of events (this is the number of "successes" in the sample).

(iii) click on options, change Confidence level to 98 (Since we have a LOS of 0.02), click on Test proportion and insert 0.3 (this is the value of the parameter under H0), make sure that Alternative says Greater than (since our H1 was p > 0.3) and check Use test and interval based on normal distribution then OK and OK again. Copy the output into your Word document.

Question 4: In step (iii), you asked Minitab to do the test and confidence interval based on normal distribution. Is that justified? Remember the condition is that np0 ≥ 10 and n(1-p0) ≥ 10, where p0 is the value of the parameter under H0. Verify if the normality assumption is justified.

{Note: If instead of the summarized data, you have the raw data of "successes" (1's) and "failures" (0's), you would do the following in step (ii):

click in the box Samples in columns, choose the column that contains your data, }

(b) Testing population mean using normal distribution approximation:

We use this approach when the population standard deviation is known or when the sample size is "large". Suppose we want to test H0 : μ = 45 against H1: μ ≠ 45., where μ is the average number of miles (in thousands of miles) a particular brand of tires will last. We take a sample of size 40 to get a sample mean of 48.3 and a sample standard deviation of 8.5. Suppose we use a level of significance of 0.01.

Question 5: Define the parameter in this problem.

Question 6: What type of test is this?

We don't know the population s.d. However, since the sample size is "large" we could use the standard normal distribution by using the sample standard deviation 8.5 as an "estimate" of the population s.d

Question 7: Carry out an appropriate hypotheses test. Based on the Minitab output, write your decision and conclusion on the hypotheses. Make sure that you copy the hypotheses to your document and also follow the hypotheses testing process.

Minitab steps:

(i) Use Stat → Basic statistics → 1-Sample Z,

(ii) Choose summarized data. {If you have the raw data in the worksheet, you will enter the corresponding column in the "Samples in columns" box.}

(iii) Enter the information for the sample size, mean and standard deviation. Note that Minitab thinks the "standard deviation" that you are supplying IS the population s.d. But you know that what you are giving is an estimate of the population s.d.)

(iv) Enter the value of the population mean under H0 in the "Hypothesized mean" box.

(v) Click "Options"; set confidence level at 99 (remember our LOS?)

(vi) Choose the "Alternative" as "Not Equal" (Remember the type of our test?)

(vii) Click OKs.

(viii) Copy the output into your Word document

Of course, a better approach is to use the t-distribution since we really don't know the population s.d.

Question 8: Carry out an appropriate t-test, write your decision and conclusion on the hypotheses. Make sure that you copy the hypotheses to your document and also follow the hypotheses testing process.

Minitab steps:

(i) Use Stat → Basic statistics → 1-sample T,

(ii) Everything else is the same as above.

(iii) Copy the output into your Word document.

Question 9: How does the p-value compare with those that you got using the standard normal distribution in Question 7?

• Save your word document as Lab04.doc.

**Lab 4 | Complete Solution**

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- Submitted On 23 Jul, 2015 06:07:35

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