STATS Lab Activity 8 – Hypothesis Testing

| August 30, 2017

Question
Lab Activity 8 – Hypothesis Testing

We will use two datasets for this activity:

For question (1),use the Fall15StudentDatadataset, which summarizes the data collected from STAT 200 students this semester in the first lesson quiz.

.gif” alt=”Text box: p-value guidelines when using standard normal table: keep this in mind: the method for finding the p-value is based on the alternative hypothesis. minitab and minitab express will provide the correct p-value if you inputted the correct alternative hypothesis. however, if doing a proportion test by hand using the standard normal table, note that the table gives you p(z < z), the cumulative probability (probability of being “less than”/area to the left). observe the following: • for ha: p ? po , the p-value =" 2•p(z"="" ?="" |z|).="" that="" is,="" find="" 1="" –="" p(z="" po, the p-value = P( Z ? z), regardless of the sign of z. • For Ha: p For question (2), use the SPXMonthlyDatadataset, which contains contains monthly total return data of S&P 500 Index since 1950.

1. Suppose that we assume the survey take of all World Campus STAT 200 students this semester is representative of all World Campus students, and we want to know if more than half of all World Campus student own a dog. UseFall15StudentDatato address this question.

a. (5 pts) What is the null hypothesis and alternative hypothesis in words and then in symbols? The null hypothesis has been done for you; please complete the alternative hypothesisfollowing the same format. Recall that an alternative hypothesis can include the symbols >, = proportion of dog owners in the sample, and then explain why this number is not enough to confidently conclude that more than half of all World Campus students own a dog.

c. Perform, by hand, the hypothesis test you defined in part a by following the steps below. Show all work.

i. (4 pts) Verify that the assumptions of the test hold (i.e., that.jpg”>.)

ii. (6 pts) Calculate the test statistic:

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iii. (4 pts) Find the p-value using the Standard Normal (Z) Table and the guidelines from the top of this lab activity.

d. (4 pts)Use softwareto confirm your test results from part c, and paste the output below.

e. (4 pts) Decide between the null hypothesis or the alternative hypothesis using ? = .05 (95% confidence level). Explain your decision using the p-value.

f. (4 pts) Write a sentence summarizing the real-world conclusion from your test, in terms of the proportion of World Campus students who are dog owners.

g. (6 pts)Suppose that we change the alternative hypothesis of the original test. For each of the following two alternative hypotheses, what would the new p-value be? You may confirm with software, but please explicitly show how you can derive this from the p-value from part cand d of the original test. Use the “P-values guidelines” at the top of this lab activity for guidance.

For Ha: p for this scenario you’ll see that it is nearly the same as the one from part b. However, your results of the test might change. Use software to re-perform the test from part a using these new numbers (see software instructions below for using Minitab/Minitab Express to do this test using the “Summarized Data” option.) Paste the output below and then answer the following questions.

i. (4 pts) What is the p-value of this test? How does it compare to the p-value obtain from part cand d?

ii. (4 pts)Does your decision and conclusion change at ? = .05? In other words, with this new p-value, would you still have made the same decision as you did in parts e and f?

iii. (4 pts) Based on what you observe, briefly explain how sample size affects the statistical significance of an observed result, even with the same sample estimate.

Note: “Statistical significance” means that the p-value is small enough to conclude the alternative hypothesis, Ha; “not significant” would mean that the p-value is relatively large and we would instead conclude the null hypothesis, H0.

2. We want to answer the question: “Does investing in S&P 500 stock index provide long-term return that is beyondthe inflation rate?” Note that the estimated average monthly inflation rate based on the Consumer Price Index (CPI) is 0.21%. Use SPXMonthlyData to address this question.

Hint:“Beyond” can be interpreted as “more than.”

Note: Though return is measured in percentage, we are NOT talking about a proportion. We are talking about a mean return, and units of return just happen to be a percentage, as does inflation rate.

a. (5 pts) What is the null hypothesis and alternative hypothesis in words and then in symbols? The null hypothesis has been done for you; please complete the alternative hypothesis. Recall that an alternative hypothesis can include the symbols >, Basic Statistics > Display Descriptive Statistics. Put “Return” in the Variable box.

Minitab Express: Statistics > Descriptive Statistics. Put “Return” in the Variable box.

Then identify the following values from the output:

Sample mean,.jpg”> =

Sample standard deviation, s =

Sample size, n =

c. Perform, by hand, the hypothesis test you defined in part a by following the steps below and using the values from part b. Use 99% confidence level (in other words, ? = .01). Show all work.

i. (2 pts) Verify that the assumptions of the test hold, i.e. that n ? 30.

ii. (6 pts) Calculate the test statistic:

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iii. (2 pts) What are the degrees of freedom for this test (DF = n-1)?

iv. (4 pts)Find the range of the p-value by using the T-distribution critical values table posted on ANGEL. To do this:

· First, find the appropriate degrees of freedom in the “df” column on the far left. If your DF is not listed, use the closest.

· Look in that row to find where your test statistic, t, from part ii, falls. It will either be between two values in that row, higher than the rightmost value, or lower than the leftmost value.

· Following your eyes up to the “Right Tail probability” critical values. These look like t with a subscript (like t.05). The subscript represents the p-values you would get for a right-tailed test (i.e., a “greater than” alternative hypothesis) with that test statistic, which is the situation we have here. Note the subscripts referring to the two values that sandwich your test statistic. Your p-value would fall between these subscripts. If your test statistic is higher than the rightmost value, your p-value is less than .001. If your test statistic is higher than the leftmost value, your p-value is greater than .100.

· For example: Suppose that DF = 20 and t= 1.45. Then, .050 < p-value < .100.

· Note that these instructions are for right-tailed tests (“greater than” alternative) only. They would have to be modified for other alternative hypotheses.

Find the p-value range from table: ____ < p-value < ____

d. (4 pts)Use softwareto confirm your test results from part c, and paste the output below.

e. (4 pts) Decide between the null hypothesis and the alternative hypothesis. Explain your decision using the p-value and ? = .01.

f. (6 pts) Write a sentence summarizing the real-world conclusion from your test, in terms of whether the mean monthly return of S&P 500 stock index is greater than the monthly rate of inflation, 0.21%.

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