# Hypothesis Testing

Question

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 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 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 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?

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