Activity 7 Hypothesis Testing – Proportion and One Mean | Complete solution

Hypothesis Testing – Proportion and One Mean

Use Dataset SPXMonthlyData.

P-value Guidelines when using Standard Normal Table (i.e. the Z-table):

Keep this in mind: The method for finding the p-value is based on the alternative hypothesis.  Minitab will provide the p-value but if doing by hand using Table A1 observe the following:

            For Ha: p ≠ p_o then the p-value = 2P(Z ≥ |z|)   That is, find 1 – P(Z < |z|) and then multiply this p-value by 2.

                For Ha: p > p_o then the p-value = P( Z ≥ z)

            For Ha: p < p_o then the p-value = P( Z ≤ z)

1. A polling group surveyed a city in Scotland regarding residents’ opinions on independence from UK. It is generally believed that the percentage of ‘Yes’ votes is 50%. The poll wants to find out whether fewer than half of the residents will vote ‘Yes’. The null hypothesis is that the percentage of ‘Yes’ votes is 0.5 (50%). The alternative hypothesis is that the ‘Yes’ vote percentage is smaller than 0.5 (50%).

a. Let p = true percentage of city residents who will vote ‘Yes’. Using mathematical notation, write null and alternative hypotheses about p.

b. The survey polled 2000 residents, of which 1050 responded that they will vote ‘Yes’ on Scotland independence.

What is the value of  p-hat = percentage of ‘Yes’ votes of the sample? How does it compare to 0.5 (the general belief)?

c.  In Minitab use Stat > Basic Stats > 1 proportion, click Summarized Data, enter 2000 for number of trials and 1050 for Number of events. Click on Options, AND enter 0.5 where it says “Test proportion.” Click on Options button. Use the default 95.0 for “Confidence level.” Select the alternative hypothesis as “Proportion < hypothesized proportion.” Choose “Normal approximation” for Method.

Minitab Express: STATISTICS>One Sample>Proportion. Under tab Data, choose Summarized data. Then enter 1050 for number of events and 2000 for number of trials. Check the box to perform hypothesis test and enter 0.5 as hypothesized proportion. Then under tab Options, choose the alternative hypothesis as Proportion< hypothesized value, use the default confidence level 95 and change the method to Normal approximation. Click OK.

What value is given for the test statistic Z in the output?      What is the p-value? 

d. Decide between the null hypothesis and the alternative hypothesis. Explain your decision.

e. Write a conclusion about how the proportion of residents in the city who will vote ‘Yes’ on Scotland independence.

 f. Suppose the study intended to find out if more than 50% of the votes will vote ‘Yes.’ In other words, the null hypothesis is the same as before but the alternative hypothesis is that the ‘Yes’ vote percentage is larger than 0.5 (50%). What is p-hat = percentage of ‘Yes’ votes of the sample. Repeat parts a, b, c, d, and e. How are the answers different from before? Explain how the alternative hypothesis affects the results of the hypothesis testing.

g. Suppose the survey had 105 ‘Yes’ responses out of 200 people (instead of 1050 out of 2000). What is the value of  p-hat = percentage of ‘Yes’ votes? How does it compare to the sample proportion for the sample used in parts b? Use software to do a hypothesis test using the null hypothesis and alternative hypothesis in part f. Decide between the null hypothesis and the alternative hypothesis. Explain your decision.

What value is given for the test statistic Z in the output?                  What is the p-value?  

h. Briefly explain how sample size affects the statistical significance of an observed result. As a starting points, note that the observed sample proportion is 0.525 for both samples in f and g, and we wish to determine if this is “significant” evidence that the true proportion is greater than 0.5. 

2. In a marketing survey for a coffee brand, 80 randomly selected coffee drinkers are asked if they only drink decaffeinated coffee. Of the 80 respondents, 7 said “yes.”

a. Let p = population proportion of coffee drinkers who only drink decaffeinated  coffee. The marketing team wants to learn if less than 10% of coffee drinkers only drink decaffeinated  coffee. Write a null and alternative hypothesis about p in this situation.  (Hint: What somebody wants to “prove” is usually the alternative.)

b. What is the value of p-hat = sample proportion that only drinks decaffeinated coffee?

c. Test the hypotheses stated in part a above. By hand, calculate the test statistic by using (Notice that this statistic is sensitive to the difference between the sample result and the null hypothesis value): 

  =  (0.0875 - .1) / sqrt(.1*(1-.1)/80) =                 (round your final value to two decimal places)

d. Use Standard Normal Table to find the p-value associated with this test statistic.  Use the p-value guidelines found at the beginning of this activity.

e. In Minitab use Stat > Basic Stats > 1 proportion, click Summarized Data, enter 80 for number of trials and 7 for Number of events. Click on Perform Hypothesis Test and enter 0.1 where it says “Hypothesized proportion” AND click Options to select the alternative hypothesis as “smaller than” AND also click on “Normal approximation” for Method.

Minitab Express: STATISTICS>One Sample>Proportion. Under tab Data, choose Summarized data. Then enter 7 for number of events and 80 for number of trials. Check the box to perform hypothesis test and enter 0.1 as hypothesized proportion. Then under tab Options, choose the alternative hypothesis as Proportion< hypothesized value, use the default confidence level 95 and change the method to Normal approximation. Click OK.

What value is given for Z in the output?        What is the p-value?  

            i. Do the Z test statistic you found by hand in part c and the p-value from part d approximately equal to the Z statistic found in part e when using Minitab?  

            ii. Decide whether the result is significant based on the p-value from the software output and report a conclusion in the context of this situation. 

            iii  What would the p-value have been if the study wanted to test if a decaffeinated coffee drinkers are exactly 10% of the coffee drinker population?  That is, test Ho: p = 0.1 versus Ha: p ≠ 0.1. (You won’t need software for this.)

3. Use SPXMonthlyData.

A financial analyst wanted answer a fundamental question faced with any investor: does investing in S&P 500 stock index provide long-term return that is beyond the inflation rate? The analyst collected monthly total return data of S&P 500 Index since 1950. She also estimated that the average monthly inflation rate based on the Consumer Price Index (CPI) is 0.21%.  Use the SPXMonthlyData file to test whether the S&P 500 monthly return is larger than average monthly inflation rate of 0.21%. Perform hypothesis testing first by hand and then with software.  The descriptive statistics are: sample size is 776; sample mean is 0.61%; and the sample standard deviation is 4.185 %.

a. Write the null and alternative hypotheses using appropriate statistical notation.

H₀:                               H_a:                 

b. Calculate DF, the t-statistic, and 95% confidence interval:

DF =               =  (0.61% - 0.21%) / (4.185% / sqrt(775)) =

c. From T-Table what is the range of the p-value based on you t-statistic?  NOTE: if you selected a two-sided Ha (i.e. used ≠) then you need to double the p-values found in the table.

d. Based on your p-value what is your decision and conclusion?

e. Now use software to verify your results. Import the dataset SPXMonthlyData into Minitab/Minitab Express.

Minitab: Go to Stat > Basic Statistics > 1-Sample t and select Return (column C8).  Click the box for “Perform Hypothesis Test” and enter the value from your hypotheses statements (i.e. u_o=0.0021).  Click on Options and select the correct alternative.  Click OK twice. 

Minitab Express: Statistics>One Sample> t. Under tab Data, select Return from the left into Sample. Check the box to perform hypothesis test and enter 0.0021 as hypothesized mean. Then under tab Options, choose mean> hypothesized value as alternative hypothesis. Use the default confidence level 95. Click OK.

Copy and paste the output here.

Do your results by hand and those from software roughly match?