Confidence Interval and Hypothesis Test | Complete Solution

Confidence Interval and Hypothesis Test:

Carbon Monoxide (CO) is a colourless and odourless gas. Even at low levels of exposure, carbon monoxide can cause serious health problems. A home is considered safe if the mean CO concentration is 5.0 parts per million (ppm) or lower. We take a random sample of 8 readings in various locations and they are recorded below:

5.8, 5.3, 5.7, 5.3, 5, 5.3, 4.9, 4.8

The mean and standard deviation for these 8 measurements are 5.262 and 0.358, respectively. Carbon monoxide levels in the home are known to follow a normal distribution.

(a) A 90% confidence interval for the true mean CO level in the house is ( ___ , ___ ). (Round values to two decimal places.)

(b) We would like to conduct a hypothesis test at the 10% level of significance to determine if the house is unsafe. What is the appropriate test statistic for this hypothesis test? Choose one below:

- z
OR
- t

(c) The hypotheses for the appropriate test of significance are, please choose one below:

- H0: u = 5.262 vs. Ha: u < 5.262
- H0: u = 5.262 vs. Ha: u > 5.262
- H0: u = 5.0 vs. Ha: u > 5.0
- H0: u = 5.0 vs. Ha: u <5.0

(d) The value of the appropriate test statistic (rounded to two decimal places) is _____.

(e) The P-value of the test is between ____ and _____ .

We record the weekly grocery bill for a random sample of 11 Canadian households. The mean and standard deviation of weekly grocery expenses for these 11 households are $114.55 and $27.76, respectively. Weekly grocery expenses are known to follow a normal distribution.

(f) A 99% confidence interval for the true mean weekly grocery bill for all Canadian households is
( ____ , ____ ). (Give the values to two decimal places.)

We would like to conduct a hypothesis test at the 1% level of significance to determine whether the true mean weekly amount spent on groceries for all Canadian households differs from $102.

(g) The hypotheses for the appropriate test of significance are, please choose one below:

- H0: u = 114.55 vs. Ha: u > 114.55
- H0: u = 102 vs. Ha: u > 102
- H0: u = 114.55 vs. Ha: u cannot equal 114.55
- H0: u = 102 vs. Ha: u cannot equal 102
- H0: X BAR = 102 vs. Ha: X BAR cannot equal 102
- H0: X = 114.55 vs. Ha: X > 114.55
- H0: X BAR = 114.55 vs. Ha: X BAR cannot equal 114.55
- H0: X BAR = 102 vs. Ha: X BAR > 102

(h) The test statistic (rounded to two decimal places) for the appropriate test of significance is _______.

(i) The P-value for the appropriate test of significance is between ____ and _____ .

(j) The correct conclusion for the appropriate test of significance is to ________ H0. Choose one below:

- Reject,
- Accept
- Fail to reject

(k) Could the confidence interval in (a) have been used to conduct the hypothesis test? Please choose one below:

- Yes. Since the value 114.55 is contained in the confidence interval, we would reject H0.
- Yes. Since the value 102 is contained in the confidence interval, we would fail to reject H0.
- Yes. Since the value 102 is not contained in the confidence interval, we would reject H0.
- Yes. Since the value 114.55 is contained in the confidence interval, we would fail to reject H0.
- Yes. Since the value 114.55 is not contained in the confidence interval, we would fail to reject H0.
- Yes. Since the value 102 is contained in the confidence interval, we would reject H0.
- Yes. Since the value 102 is not contained in the confidence interval, we would fail to reject H0.
- Yes. Since the value 114.55 is not contained in the confidence interval, we would reject H0.
- No, the confidence interval could not be used to conduct the test.

Matched Pairs:

We would like to conduct a hypothesis test at the 5% level of significance to determine whether husbands are older than their wives on average. The ages of a random sample of 20 married couples are recorded. Some summary statistics that may be helpful are shown in the table below:

Husband Wife Difference (H - W)
Sample Mean 43.59 41.24 2.35
Sample Std. Dev. 8.09 6.67 3.1

(l) What are the hypothese for the appropriate test of significance? Choose one below:

- H0: X BAR d = 0 vs. Ha: X BAR d < 0
- H0: X BAR H = X BAR W vs. Ha: X BAR H < X BAR W
- H0: X BAR d = 0 vs. Ha: X BAR d > 0
- H0: uH = uW vs. Ha: uH > uW
- H0: X BAR H = X BAR W vs. Ha: X BAR H > X BAR W
- H0: uH = uW vs. Ha: uH < uW
- H0: ud = 0 vs. Ha: ud > 0
- H0: ud = 0 vs. Ha: ud < 0

(m) The test statistic (rounded to two decimal places) for the appropriate test of significance is
t = _____.

(n) The P-value of the test is between ____ and _____.

(o) What is the correct conclusion of the test? Choose one below:

- Reject H0. There is sufficient evidence that husbands are older than their wives on average.
- Reject H0. There is insufficient evidence that husbands are older than their wives on average.
- Fail to reject H0. There is insufficient evidence that husbands are older than their wives on average.
- Fail to reject H0. There is sufficient evidence that husbands are older than their wives on average.

Sample Size for Proportions:

We would like to estimate the true proportion of high school students who own a cell phone.

(p) What sample size is required to estimate this proportion to within 0.038 with 89% confidence? _____

(q) What sample size is required to estimate this proportion to within 0.019 with 89% confidence? _____

(r) When we decrease the desired margin of error to half of its original value, we require _____ times the sample size.

(s) Using only your answer from (a), what sample size is required to estimate this proportion to within 0.057 with 89% confidence? _____

(t) Suppose we believe that about 83.6% of high school students own a cell phone. What sample size is required to estimate the true proportion to within 0.024 with 96% confidence? _____

Inference for Proportions:

The Canadian Senate is described by critics as "unelected and unaccountable". Given recent scandals in the upper chamber, more and more Canadians are calling for the abolition of the Senate.

In a random sample of 354 Canadians, 196 said they support the abolition of the Senate.

(u) A 95% confidence interval for the true proportion of Canadians who favour the abolition of the Senate is ( ____ , ____ ). (Round your answers to four decimal places.)

(v) We would like to conduct a hypothesis test at the 5% level of significance to determine whether there is evidence that the majority of Canadians support the abolition of the Senate. The hypotheses for the appropriate test of significance are, choose one below:

- H0: p = 0.5 vs. Ha: p < 0.5
- H0: p = 0.5 vs. Ha: p > 0.5
- H0: p = 0.5 vs. Ha: p cannot equal 0.5
- H0: p hat = 0.5 vs. Ha: p hat < 0.5
- H0: p hat = 0.5 vs. Ha: p hat cannot equal 0.5
- H0: p hat = 0.5 vs. Ha: p hat > 0.5

(w) The test statistic (rounded to two decimal places) for the appropriate test of significance is z = ______

(x) The P-value of the test is ______ .

(y) The correct conclusion of the test is to _____ H0. Choose one below:

- Reject
- Accept
- Fail to reject

In the 2011 Canadian general election, the Conservative Party received 39.62% of all votes cast. In a random sample of 370 Canadian voters this year, 128 of them (34.59%) said they support the Conservative Party.

We would like to conduct a hypothesis test at the 10% level of significance to determine whether there is evidence that support for the Conservative Party has changed since the 2011 election.

(1) The hypotheses for the appropriate test of significance are, choose one below:

- H0: p = 0.3459 vs. Ha: p < 0.3459
- H0: p = 0.3459 vs. Ha: p > 0.3459
- H0: p = 0.3962 vs. Ha: p < 0.3962
- H0: p = 0.3962 vs. Ha: p > 0.3962
- H0: p = 0.3962 vs. Ha: p cannot equal 0.3962
- H0: p = 0.3459 vs. Ha: p cannot equal 0.3459

(2) The test statistic (rounded to two decimal places) for the appropriate test of significance is
z = ______ .

(3) The P-value of the test is ______ .

(4) The correct conclusion of the test is to ______ H0. Choose one below:

- Reject
- Accept
- Fail to reject