Problem 1.
An internet service provider (ISP) provides internet connections to 100,000 customers.
10,000 of the customers have high-speed connections and 90,000 of the customers have low-speed connections.
The ISP wants to know whether, on the average, customers who have high-speed connections use email more frequently than customers who have low-speed connections.
To find out, the ISP takes a simple random sample of 300 high-speed-connection customers and an independent random sample of 400 low-speed-connection customers.
For each customer in the sample, they find the number of email messages sent and received in the previous month.
Then they compute the sample mean and sample standard deviation of each of the two sets of numbers.
Let h denote the average number of emails sent and received in the previous month among all customers with high-speed connections, and let l denote the average number of emails sent and received in the previous month among all customers with low-speed connections.
Let H and L be the corresponding sample means.
Let Sh be the sample standard deviation of the number of emails sent and received in the previous month by customers with high-speed connections, and let Sl be the sample standard deviation of the number of emails sent and received in the previous month by customers with low-speed connections.
Which of these hypotheses is the most appropriate null hypothesis for this problem? (Q1)
A: h > l
B: H > L
C: h < l
D: H = L
E: H < L
F: h = l
Which of these hypotheses is the most appropriate alternative hypothesis for this problem? (Q2)
A: h = l
B: H > L
C: H = L
D: H < L
E: h < l
F: h > l
Suppose that the population distributions of the number of emails sent in the by high-speed-connection customers and by low-speed-connection customers both are nearly normal.
Which of the following have probability histograms that can be approximated well by a normal curve, after transforming to standard units? (select all that apply)
(Q3)
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Question
An internet service provider (ISP) provides internet connections to 100,000 customers. 10,000 of the customers have high-speed connections and 90,000 of the customers have low-speed connections.
The ISP wants to know whether, on the average, customers who have high-speed connections use email more frequently than customers who have low-speed connections.
To find out, the ISP takes a simple random sample of 150 high-speed-connection customers and an independent random sample of 300 low-speed-connection customers.
For each customer in the sample, they find the number of email messages sent and received in the previous month. Then they compute the sample mean and sample standard deviation of each of the two sets of numbers.
Let h denote the average number of emails sent and received in the previous month among all customers with high-speed connections, and let l denote the average number of emails sent and received in the previous month among all customers with low-speed connections.
Let H and L be the corresponding sample means.
Let Sh be the sample standard deviation of the number of emails sent and received in the previous month by customers with high-speed connections, and let Sl be the sample standard deviation of the number of emails sent and received in the previous month by customers with low-speed connections.
(Q1) Which of these hypotheses is the most appropriate null hypothesis for this problem?
A: h > l
B: H > L
C: h < l
D: H = L
E: H < L
F: h = l
(Q2) Which of these hypotheses is the most appropriate alternative hypothesis for this problem?
A: h = l
B: H > L
C: H = L
D: H < L
E: h < l
F: h > l
(Q3) Suppose that the population distributions of the number of emails sent in the by high-speed-connection customers and by low-speed-connection customers both are nearly normal.
Which of the following have probability histograms that can be approximated well by a normal curve, after transforming to standard units? (select all that apply)
A: L
B: H-L
C: H
D: I
E: h - l
F: h
Suppose we construct a Z statistic by transforming H - L to standard units (approximately).
Under the alternative hypothesis, the expected value of Z would be (Q4)
A: Negative
B: zero
C: Positive
, so we should (Q5)
A: consult a statistician
B: use a left-tail test
C: use a right-tail test
D: use a two-tail test
To test the null hypothesis at significance level 10%, we should reject the null hypothesis if (Q6)
A: the z-score
B: the absolute value of the z-score
(Q7)
A: less than
B: greater than
(Q8)
____? (continues from q6 and q7)
For high-speed-connection customers, the sample mean number of emails in the month is 293, and the sample standard deviation of the number of emails in the the month is 117. For low-speed-connection customers, the sample mean number of emails in the month is 274, and the sample standard deviation of the number of emails in the month is 135.
The estimated standard error of H - L is (Q9)
The z-score is (Q10)
The P-value of the null hypothesis is (Q11)
The ISP should reject the null hypothesis. (true/false) (Q12)
Problem 2.
At a particular university in an urban area, official policy mandates that university-owned student housing shall rent for no more than 80% of the market rate for comparable housing.
Rent for all two-bedroom university-owned student apartments is $570/month. All the university-owned two-bedroom student apartments have one bathroom, no view, and are comparable in construction, size, age, amenities, etc.
To determine whether the rent satisfies the rules, an administrator proposes to compile as complete a list as he can of two-bedroom apartments for rent in the area, using sources including newspaper ads, commercial rental listing services, and bulletin boards.
Then, he will take a simple random sample of 150 of the apartments in the list, and visit each one to determine whether it is comparable to the university-owned apartments in size, number of bathrooms, state of repair, amenities (such as laundry facilities, bathtub/shower), etc.
He will compute the sample mean rent of those apartments he finds to be comparable to the university-owned apartments. Let r denote the mean rent of all comparable two-bedroom apartments in the area, and let R denote the sample mean rent of the comparable two-bedroom apartments the administrator finds.
The administrator will approach the problem of determining whether the university is complying with the mandate as an hypothesis test.
The most appropriate alternative hypothesis is (Q13) r or R (Q14) <, =, or >? (Q15)$____? .
Suppose that the administrator finds that 28 of the apartments are comparable to two-bedroom university-owned student apartments.
Assume that these 28 apartments can be treated as a random sample of size 28 with replacement from the population of comparable two-bedroom apartments for rent in the area, and the distribution of rents for comparable apartments in the area is approximately normal.
Suppose that the sample mean of the rents is $681 and the sample standard deviation of the rents is $84.
The estimated standard error of the sample mean is $ (Q16)
The number of degrees of freedom for Student's t-curve to approximate the probability histogram of the T statistic is (Q17)
The observed value of the T statistic is (Q18)
The P-value of the null hypothesis is (Q19)
The null hypothesis should be rejected at significance level 5%. (Q20)
A (two-sided) 95% confidence interval for the mean rent of comparable two-bedroom apartments in the area is from $ (Q21) (low) to $ (Q22) (high).