**Hypothesis Testing – Comparing Two Groups**

**1**For each of the following research questions does
the situation or research question involve independent samples or
paired data?

** **

**a.**Do liberal arts majors and science majors have
different SAT verbal scores, on average?

** **

** **

**b.**Does a caffeine pill help Autistic patients
score higher on a cognitive exam?

** **

** **

**c.**In heterosexual relationships, does the man or
the woman average more time outside the house?

d. Do families with at least one unemployed adult average more time watching TV per week than families in which all adults are employed?

**2 **In the
**Datasets folder**open the
**GSS Dataset**. The data are from the
2006 General Social Survey, a federally funded national survey done
every other year by the University of Chicago. The variable
**
marital
**indicates whether the respondent is presently married or
not. We’ll compare the

**mean**age (

**is the variable) for those who are married versus those who are not.**

*Age*

**a.**In words, write a null hypothesis for this
situation. We’re comparing two means (age for married people versus
unmarried people).

**b.**Using statistical notation for means write null
and alternative hypotheses for this problem.

**c.**Recall from the lecture notes that when doing a
two-sample t-test one consideration is whether the two standard
deviations (or variances) are equal. To check, use software
to find the standard deviation for
**Age**for the two categories of
**marital**status. In Minitab Express, you can
get these SD by clicking Statistics > Descriptive Statistics and
selecting Age for Variables and marital for Group
variable.

**
i. **What
are the two standard deviations?

Std.dev for married:

Std.dev for not married:

**ii.**Is the larger standard deviation
**more than**twice the smaller standard deviation?

**iii.**If your answer to part
**ii**is “Yes” then we will use the
*unpooled*method for calculating the standard error.
If your answer was “No” then we can use the
*pooled*method. Which method should we use?

**d.**The two-sample t-test is used to compare means
when data is from two independent samples (as it is here). Use
software to conduct a
**2-sample t test**. If your answer to
**part iii**above is “NO” use
*pooled*then click the box for “Assume Equal Variances”.
Read the output to find the values of the
*t*-statistic and the p-value.

t= p-value =

e. State a conclusion about the hypotheses and about the “real world” situation.

f. The formula for the pooled
*t*-statistic is
. Give values
for each of the elements in the formula.

**g.**The output includes a 95% confidence interval
for the difference between means. Write a sentence that interprets
this interval in terms of how much difference there is between the
mean ages of the two groups.

**h.**Refer again the to the 95% confidence interval
of the previous part. Explain why it is evidence that makes it
reasonable to conclude that the population means differ.

**3**In a national survey of 12
^{th}graders, 254 of 1356 boys said they never or rarely
wear a seatbelt when driving. Among 1168 girls, 197 said they never
or rarely wear a seatbelt when driving.

**a.**Let p
_{1}= population proportion that never or rarely wears a
seatbelt for boys and p
_{2}= the corresponding proportion for girls. Write null
and alternative hypotheses about p
_{1}and p
_{2}for testing if a difference exists.

** **

**b.**
**If using Minitab Express**, then use
**Statistics > Two Samples >
Proportions**. Click on the drop-down menu under the
word “Data” and select
*Summarized data*. Use the boys as the first sample and
girls as the second sample. “Number of trials” means sample size
and “Number of events” means number rarely or never wearing a
seatbelt. Next click
**Options**and select your alternative hypothesis and
enter your test difference.
** **If your hypotheses statements are testing
that the difference is 0, then be sure to select the
*Use the pooled estimate of the proportion*for your test
method
*. *Use the output to give values for the
following:

For boys, sample proportion = _______ For girls, sample proportion = = ________

The difference between the sample proportions is =_______

Value of test statistic = _________ p-value = _______

**c.**Explain whether we can we say there is a
difference between the population proportions in this
situation.

**4** In the
**Datasets folder**click the link for the
**Class Survey**. Is there a difference in the
number of parties attended each month and the number of days
drinking alcohol? Since we are considering differences
between two measurements (i.e. Monthly Parties and Drinking Days)
on the same individual we can consider the data to be paired.
Use software to conduct a
**Matched Pairs t-test**.

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

H
_{
0}:
H
_{
a}:

** **

**b.**Based on the Confidence interval what is your
conclusion?

**c.** Based on your p-value what is your
conclusion?

**d.**Use the data from the output to calculate the
t-statistic by:
*t*
**=**
**= **

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