Question 1

The number of trials and the population proportion are respectively represented by what symbols?
A)p and n
B)α and β
C)z and t
D)n and π

Accepted Answer

"n" represents the number of trials and "π" represents the population proportion in a binomial distribution. Therefore, option D is the correct answer. Option A is incorrect because "p" is often used to represent the sample proportion in statistics. Option B is incorrect because alpha (α) and beta (β) are used to represent the levels of significance and power in hypothesis testing. Option C is incorrect because "z" and "t" are used to represent test statistics in hypothesis testing.

Question 2

An electronics retailer believes that,at most,40% of their cell phone inventory was sold during November. The retailer surveyed 80 dealers and found that 38% of the inventory was sold. Since 38% is less than 40%,is this difference of 2 percentage points sampling error or a significant difference? Test at the 0.05 level. The computed z = −0.37.
A)The 2% is a significant difference.
B)We cannot determine if the 2% is a significant difference.
C)There is not enough information to reach a conclusion.
D)None of the other answers apply.

Accepted Answer

The computed z-score of −0.37 is not significant enough to reject the null hypothesis that the proportion sold is equal to or less than 40%. Therefore, the difference of 2 percentage points is likely just due to sampling error and not a significant difference.

Question 3

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the z-test statistic?
A)1.02
B)1.22
C)−1.02
D)−1.22

Accepted Answer

The z-test statistic is calculated using the formula $z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$, where $\hat{p}$ is the sample proportion, $p$ is the population proportion, and $n$ is the sample size. Here, $\hat{p} = 0.36$, $p = 0.41$, and $n = 100$. Plugging these values into the formula gives $z = \frac{0.36 - 0.41}{\sqrt{\frac{0.41(1-0.41)}{100}}} = -1.02$.

Question 4

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. If α = 0.025,what will be the decision?
A)Accept the null.
B)Reject the null and conclude the defects are not greater than 10%.
C)Reject the null and conclude the defects are greater than 10%.
D)Fail to reject the null.

Accepted Answer

The sample proportion of defective peaches is 50/400 = 0.125. The null hypothesis is that the proportion of defective peaches is less than 0.10. The alternative hypothesis is that the proportion of defective peaches is greater than 0.10. The test statistic is z = (0.125 - 0.10) / sqrt(0.10 * 0.90 / 400) = 2.5. The p-value is P(Z > 2.5) = 0.0062. Since the p-value is less than 0.025, we reject the null hypothesis and conclude that the proportion of defective peaches is greater than 0.10.

Question 5

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion?
A)0.10
B)0.125
C)40
D)0.40

Accepted Answer

The sample proportion is calculated by dividing the number of defective peaches by the total number of peaches examined. So, 50 defective peaches out of 400 gives a proportion of 50/400 = 0.125.

Question 6

To test the hypothesis that 55% of those families who plan to purchase a vacation residence in Florida want a condominium,the null hypothesis is π = 0.55 and the alternate is π ≠ 0.55. A random sample of 400 families who planned to buy a vacation residence revealed that 228 families want a condominium. What decision should be made regarding the null hypothesis using the 0.01 level of significance?
A)Do not reject it.
B)Reject it.
C)Cannot accept it or reject it based on the information given.
D)Purchase a condominium.

Accepted Answer

The sample proportion of families who want a condominium is 228/400 = 0.57. The standard error of the proportion is sqrt(0.55*0.45/400) = 0.024. The test statistic is (0.57 - 0.55)/0.024 = 0.833. The p-value is P(|Z| > 0.833) = 2*P(Z > 0.833) = 2*0.2033 = 0.4066. Since the p-value is greater than 0.01, we do not reject the null hypothesis.

Question 7

If 20 out of 50 students sampled live in a college dormitory,what is the estimated proportion of students at the university living in a dormitory?
A)0.20
B)0.40
C)0.50
D)0.60

Accepted Answer

The proportion of students living in a dormitory can be estimated by dividing the number of students living in a dormitory by the total number of students sampled:
20/50 = 0.4
Therefore, the estimated proportion of students at the university living in a dormitory is 0.40 or 40%, which corresponds to choice B.

Question 8

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for α = 0.025?
A)+1.960
B)±1.645
C)−1.960
D)−1.645

Accepted Answer

For a one-tailed test with α = 0.025, the critical value is -1.960.

Question 9

The claim that "40% of those persons who retired from an industrial job before the age of 60 would return to work if a suitable job was available" is to be investigated at the 0.02 significance level. If 74 out of the 200 workers sampled said they would return to work,what is our decision?
A)Do not reject the null hypothesis because −0.866 lies in the region between +2.326 and −2.326.
B)Do not reject the null hypothesis because −0.866 lies in the region between +2.576 and −2.576.
C)Reject the null hypothesis because 37% is less than 40%.
D)Do not reject the null hypothesis because 37% lies in the area between 0% and 40%.

Accepted Answer

The decision is based on the z-score calculation and its comparison to the critical value at a 0.02 significance level. The calculated z-score of -0.866 does not exceed the critical z-values of ±2.326, indicating that the observed proportion (37%) does not significantly differ from the hypothesized proportion (40%) at the 0.02 significance level.

Question 10

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is your decision if α = 0.01?
A)Fail to reject the null hypothesis.
B)Reject the null hypothesis and conclude the newscast does not reach 41% of the audience.
C)Fail to reject the alternate and conclude the newscast does not reach 41% of the audience.
D)Reject the alternate and conclude the newscast reaches about 41% of the audience.

Accepted Answer

Given the information, we do not have sufficient evidence to reject the null hypothesis that the newscast reaches 41% of the audience. The sample percentage (36%) is close to the claimed percentage (41%), and without conducting a formal hypothesis test here, we typically fail to reject the null hypothesis when the sample estimate is reasonably close to the population parameter being tested, especially considering a stringent significance level of α = 0.01.

Question 11

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the critical value if the level of significance is 0.10?
A)−1.282
B)±1.645
C)−2.576
D)+2.576

Accepted Answer

To find the critical value, we need to use a z-test. We can use the formula z = (p̂ - p) / √(p(1-p) / n), where p̂ is the sample proportion, p is the claimed proportion, n is the sample size, and we assume a two-tailed test since we don't have a specific direction. Setting α = 0.10, we find the z-value that corresponds to the area between -0.05 and 0.05 (the middle 90% of the distribution). Using a standard normal table or calculator, we find that z = ±1.645. Therefore, the critical value is ±1.645.

Question 12

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the z-test statistic?
A)+0.125
B)+0.278
C)−1.645
D)+1.667

Accepted Answer

The z-test statistic is calculated using the formula $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size. Here, $\hat{p} = \frac{50}{400} = 0.125$, $p_0 = 0.10$, and $n = 400$. Plugging these values into the formula gives $z = \frac{0.125 - 0.10}{\sqrt{\frac{0.10(1-0.10)}{400}}} = +1.667$.

Question 13

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the alternate hypothesis for a one-sided test? A)H₁: π ≥ 0.10 B)H₁: π > 0.10 C)H₁: π ≤ 0.10 D)H₁: π < 0.10

Accepted Answer

The alternate hypothesis for a one-sided test in this case would be that the proportion, π, of defective peaches in a bushel is less than 10%. Therefore, the correct choice is D, which states H₁: π < 0.10.

Question 14

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the critical value if α = 0.01?
A)+2.576
B)+2.326
C)±2.576
D)−2.326

Accepted Answer

To determine the critical value we need to perform a hypothesis test for the proportion.

Null hypothesis: p=0.41
Alternative hypothesis: p<0.41 (since we are looking for the critical value at alpha=0.01)

We can use the normal distribution to find the critical value, since the sample size (100) is large enough to satisfy the conditions of the central limit theorem.

Using a standard normal distribution table, the critical value for a left-tailed test at alpha=0.01 is approximately -2.576.

Therefore, the answer is C (±2.576), since we are looking for the critical value in the left-tail of the distribution.

Question 15

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the null hypothesis? A)H₀: π = 0.36 B)H₀: π = 0.41 C)H₀: π ≠ 0.36 D)H₀: µ = 0.41

Accepted Answer

The null hypothesis is always a statement of no difference or no effect. In this case, the null hypothesis is that the proportion of viewers who watch the late evening news on this local CBS station is equal to the claimed reach of 41% (π = 0.41), which is what the local CBS affiliate claims. Therefore, the correct null hypothesis is H₀: π = 0.41. Option B is the only one that states this.

Question 16

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the alternate hypothesis? A)H₁: π = 0.36 B)H₁: π = 0.41 C)H₁: π ≠ 0.41 D)H₁: µ ≠ 0.41

Accepted Answer

The alternate hypothesis should reflect the belief that the sample proportion of viewers who watch the late evening news on the local CBS station is different from the claimed reach of the 11 p.m. newscast by the station, which is 41%. Therefore, option C, H₁: π ≠ 0.41 is the best choice. Options A and B reflect the null hypothesis, while option D is not relevant as it refers to a mean rather than a proportion.

Question 17

A sample of 250 adults tried the new multigrain cereal Wow! A total of 187 rated it as excellent. In a sample of 100 children,66 rated it as excellent. Using the 0.1 significance level,the researcher wishes to show that adults like the cereal better than children. What test statistic should we use to compare the ratings of adults and children?
A)A z-statistic
B)A right one-tailed test statistic
C)A left one-tailed test statistic
D)A t-statistic

Accepted Answer

A z-statistic should be used because the sample sizes are relatively large for both the adults (n = 250) and the children (n = 100), and the researcher is comparing two independent groups. Additionally, the question is asking for a one-tailed test (specifically, to test whether adults like the cereal better than children), which means we need to use a z-statistic to find the probability of getting a difference as extreme as the one observed assuming the null hypothesis is true. A t-statistic would be appropriate if the sample sizes were smaller or if the population variances were unknown, but neither of those conditions apply here.

Question 18

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the sample proportion?
A)0.41
B)0.36%
C)0.41%
D)0.36

Accepted Answer

The question is asking for the sample proportion, which is the percentage of viewers in the survey who indicated that they watch the late evening news on the local CBS station. This is given as 36%, so the answer is D, 0.36. A is incorrect because it refers to the percentage of the overall viewing audience, not the sample surveyed. B is incorrect because it uses a percentage sign and should be a decimal. C is incorrect for the same reason as A.

Question 19

The sample proportion is defined as ________.
A)nπ
B)x/n
C)n!
D)π

Accepted Answer

The sample proportion, denoted as p̂, is defined as the number of successes in the sample (x) divided by the sample size (n). Therefore, the formula for the sample proportion is p̂ = x/n, which corresponds to option B in the given choices. Option A (nπ) does not correspond to the formula for sample proportion. Option C (n!) is the formula for factorial, which is not related to sample proportion. Option D (π) refers to the mathematical constant pi, which is not related to sample proportion.

Question 20

It is claimed that in a bushel of peaches,less than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the null hypothesis? A)H₀: π ≠ 0.10 B)H₀: π ≥ 0.10 C)H₀: π ≤ 0.10 D)H₀: π < 0.10

Accepted Answer

The null hypothesis is always the opposite of what is being claimed or tested. Here, the claim is that less than 10% (0.10) are defective, so the null hypothesis would be that the proportion of defective peaches is equal to or greater than 10% (0.10). Therefore, choice B is the correct option, which states the null hypothesis as H₀: π ≥ 0.10.

Quiz 15: Nonparametric Methods: Nominal Level Hypothesis Tests

The number of trials and the population proportion are respectively represented by what symbols?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the z-test statistic?

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. If α = 0.025,what will be the decision?

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the sample proportion?

If 20 out of 50 students sampled live in a college dormitory,what is the estimated proportion of students at the university living in a dormitory?

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the critical value for α = 0.025?

The claim that "40% of those persons who retired from an industrial job before the age of 60 would return to work if a suitable job was available" is to be investigated at the 0.02 significance level. If 74 out of the 200 workers sampled said they would return to work,what is our decision?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is your decision if α = 0.01?

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the z-test statistic?

It is claimed that in a bushel of peaches,fewer than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the alternate hypothesis for a one-sided test?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the critical value if α = 0.01?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the null hypothesis?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the alternate hypothesis?

Based on the Nielsen ratings,the local CBS affiliate claims its 11 p.m. newscast reaches 41% of the viewing audience in the area. In a survey of 100 viewers,36% indicated that they watch the late evening news on this local CBS station. What is the sample proportion?

The sample proportion is defined as ________.

It is claimed that in a bushel of peaches,less than 10% are defective. A sample of 400 peaches is examined and 50 are found to be defective. What is the null hypothesis?