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Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top $30$ stocks of banks in the northeast and in the west as identified by Nasdaq on May $24,2013$ are listed in Table $10.6$ and Table$10.7,$ respectively

Is there a difference in the weighted alpha of the top $30$ stocks of banks in the northeast and in the west? Test at a $5\%$ significance level. Answer the following questions:

a. Is this a test of two means or two proportions?

b. Are the population standard deviations known or unknown?

c. Which distribution do you use to perform the test?

d. What is the random variable?

e. What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.

f. Is this test right, left, or two tailed?

g. What is the p-value?

h. Do you reject or not reject the null hypothesis?

i. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.

j. Calculate Cohen’s d and interpret it

Researchers conducted a study to find out if there is a difference in the use of eReaders by different age groups. Randomly selected participants were divided into two age groups. In the $16$- to $29$-year-old group, $7\%$of the $628$surveyed use eReaders, while of the $2,309$participants $30$years old and older use eReaders.

Adults aged 18 years old and older were randomly selected for a survey on obesity. Adults are considered obese if their body mass index (BMI) is at least $30$. The researchers wanted to determine if the proportion of women who are obese in the south is less than the proportion of southern men who are obese. The results are shown in Table 10.27. Test at the 1% level of significance.

Two computer users were discussing tablet computers. A higher proportion of people ages $16$to $29$use tablets than the proportion of people age $30$and older. Table localid="1653641949025" $10.28$details the number of tablet owners for each age group. Test at the localid="1653641953436" $1\%$level of significance.

Table.localid="1653641957518" $10.28$

Two types of valves are being tested to determine if there is a difference in pressure tolerances. Fifteen out of a random sample of $100$of Valve $A$cracked under $4500\text{psi.}$Six out of a random sample of $100$of Valve $B$cracked under $4500\text{psi}.$Test at a$5\%$ level of significance.

A group of friends debated whether more men use smartphones than women. They consulted a research study of smartphone use among adults. The results of the survey indicate that of the $973$men randomly sampled, $379$use smartphones. For women,$404$ of the $1,304$ who were randomly sampled use smartphones. Test at the $5\%$ level of significance.

It is thought that teenagers sleep more than adults on average. A study is done to verify this. A sample of 16 teenagers has a mean of 8.9 hours slept and a standard deviation of 1.2. A sample of 12 adults has a mean of 6.9 hours slept and a standard deviation of 0.6.

Use the data sets found in Appendix C to answer this exercise. Is the proportion of race laps Terri completes slower than $130$ seconds less than the proportion of practice laps she completes slower than $135$ seconds?

Ten individuals went on a low–fat diet for 12 weeks to lower their cholesterol. The data are recorded in Table 10.30. Do you think that their cholesterol levels were significantly lowered?

$\begin{array}{|ll|}\hline \text{Starting cholesterol level}& \text{Ending cholesterol level}\\ 140& 140\\ 220& 230\\ 110& 120\\ 240& 220\\ 200& 190\\ 180& 150\\ 190& 200\\ 360& 300\\ 280& 300\\ 260& 240\\ \hline\end{array}$

Use the following information to answer the next two exercises. A new AIDS prevention drug was tried on a group of 224 HIV positive patients. Forty-five patients developed AIDS after four years. In a control group of 224 HIV positive patients, 68 developed AIDS after four years. We want to test whether the method of treatment reduces the proportion of patients that develop AIDS after four years or if the proportions of the treated group and the untreated group stay the same.

Let the subscript $t=$treated patient and $ut=$untreated patient.

If the p-value is 0.0062 what is the conclusion (use α = 0.05)?

a. The method has no effect.

b. There is sufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years.

c. There is sufficient evidence to conclude that the method increases the proportion of HIV positive patients who develop AIDS after four years.

d. There is insufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years.