STATS Lab Activity 07 Assignment

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. H0: p=.50 versus Ha: p < .50
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)? 1050/2000= .52 Sample value is close to .50
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.
SPSS Users: Open the Excel Summarized Procedures and select the tab “Z test of a Proportion”. Enter
0.525 as the Sample Proportion; 2000 for the Sample Size; and 0.5 for the Hypothesized Proportion.
NOTE: The resulting p-value is for a two-sided test (i.e. “not equal” alternative hypothesis – if you are
conducting a one-sided test, where your alternative is specified as either “less than” or “greater than” you
will need to cut this p-value in half to arrive at the proper p-value for the one-sided test).

What value is given for the test statistic Z in the output? 2.24

What is the p-value? .9873

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 Minitab 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 pvalue 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.
SPSS Users: Open the Excel Summarized Procedures and select the tab “Z test of a Proportion”. Enter
0.0875 as the Sample Proportion; 80 for the Sample Size; and 0.1 for the Hypothesized Proportion.
NOTE: The resulting p-value is for a two-sided test (i.e. “not equal” alternative hypothesis – if you are
conducting a one-sided test, where your alternative is specified as either “less than” or “greater than” you
will need to cut this p-value in half to arrive at the proper p-value for the one-sided test).

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 Minitab 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

3 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 Minitab.
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:

Ha:

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 Minitab to verify your results. 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. uo). Click on Options and select the correct alternative. Click OK
twice and copy and paste your Minitab results. Do your results by hand and those from Minitab
roughly match?