Hypothesis Testing – Proportion and One Mean

Hypothesis Testing – Proportion and One Mean

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 ≠ po 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 > po then the p-value = P( Z ≥ z)

For Ha: p < po 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 940 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 Express use Statistics > One Sample > Proportion, click the first drop-down menu underneath the word “Data”, and select Summarized Data. Enter 940 for “Number of events”, and 2000 for “Number of trials”. Check the box that says “Perform hypothesis test”, and enter 0.50 in the box labeled “Hypothesized proportion:”. Click on Options and use the default 95 for “Confidence level.” Select the alternative hypothesis as “Proportion < hypothesized proportion.” Choose “Normal approximation” for Method. Paste your output below.

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 s is that the ‘Yes’ vote percentage is larger than 0.5 (50%). What is p-hat = percentage the alternative hypothesis of ‘Yes’ votes of the sample. Repeat parts a, b, c, d, and e. You'll need to change one of the options when performing the hypothesis test in Minitab Express. Paste the output below. How are the answers different from before? Explain how the alternative hypothesis affects the results of the hypothesis testing.

g. Suppose the survey had 94 ‘Yes’ responses out of 200 people (instead of 940 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 Minitab Express 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 0.470 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, 120 randomly selected coffee drinkers are asked if they only drink decaffeinated coffee. Of the 120 respondents, 16 said “yes.”

a. Let p = population proportion of coffee drinkers who only drink decaffeinated coffee. The marketing team wants to learn if less than 15% 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):

= (0.1333 - .15) / sqrt(.15*(1-.15)/120) = (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 Express use Statistics > One Sample > Proportion, click the first drop-down menu underneath the word “Data”, and select Summarized Data. Enter 16 for “Number of events”, and 120 for “Number of trials”. Check the box that says “Perform hypothesis test”, and enter 0.15 in the box labeled “Hypothesized proportion:”. Click on Options and use the default 95 for “Confidence level.” Select the alternative hypothesis as “Proportion < hypothesized proportion.” Choose “Normal approximation” for Method. Paste your output below.

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 Express?

ii. Decide whether the result is significant based on the p-value from Minitab Express 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 15% of the coffee drinker population? That is, test Ho: p = 0.15 versus Ha: p ≠ 0.15

3

I want to know if the average height of male PSU students is greater than the national average of 69.7 inches. Use the MenHeights data set, which is a small sample of PSU male students. Perform hypothesis testing first by hand and then with Minitab Express. The descriptive statistics are: sample size is 99; sample mean is 70.2424; and the sample standard deviation is 3.6340.

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

H0: Ha:

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

DF = = (70.2424 – 69.7) / (3.6340 / sqrt(99)) =

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 Minitab Express to verify your results. Go to Statistics > One Sample > t and select MenHeights. Click the box for “Perform Hypothesis Test” and enter the value from your hypotheses statements (i.e. uo). Click on Options and select the correct alternative. Click OK twice and copy and paste your output. Do your results by hand and those from Minitab Express roughly match?