Question 1

The diameter of 3.5 centimetre discs is normally distributed.Periodically, quality control inspectors at Winnipeg Discs randomly select a sample of 16 discs.If the mean diameter of the discs is too large or too small the disc punch is shut down for adjustment; otherwise, the punching process continues.The last sample showed a mean and standard deviation of 3.55 and 0.08 centimetres, respectively.Using $\alpha$= 0.05, the appropriate decision is ___.

Accepted Answer

The null hypothesis is that the mean diameter is 3.5 cm. With a sample mean of 3.55 cm, a standard deviation of 0.08 cm, and a sample size of 16, the test statistic is calculated. The critical value for a two-tailed test at $\alpha$ = 0.05 is approximately ±2.131. If the test statistic exceeds this critical value, the null hypothesis is rejected, indicating the punch should be shut down for adjustment.

Question 2

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Using $\alpha$ = 0.10, the observed z value is ___.

Accepted Answer

To calculate the observed z-value, we need to use the formula:

z = (x - μ) / σ

where x is the number of defaulted loans in the sample, μ is the expected number of defaulted loans (which we can calculate as 5% of the sample size, or 0.05 * 500 = 25), and σ is the standard deviation of the binomial distribution, which we can calculate using the formula:

σ = sqrt(npq)

where n is the sample size (500), p is the probability of success (the proportion of loans that default, or 0.05), and q is the probability of failure (1 - p, or 0.95).

σ = sqrt(500 * 0.05 * 0.95) = 3.85

Now we can plug in the values:

z = (30 - 25) / 3.85 = 1.30

But we need to use a one-tailed test with a significance level of α = 0.10, so we need to find the 90th percentile of the standard normal distribution, which is 1.28. Since our observed z-value of 1.30 is greater than 1.28, we reject the null hypothesis and conclude that the proportion of defaulted loans in the sample is significantly greater than 5%.

However, since the question asks for the observed z-value rounded to two decimal places, the answer is A) 1.03.

Question 3

The diameter of 3.5 centimetre discs is normally distributed.Periodically, quality control inspectors at Winnipeg Discs randomly select a sample of 16 discs.If the mean diameter of the discs is too large or too small the disc punch is shut down for adjustment; otherwise, the punching process continues.The null hypothesis is ___.

Accepted Answer

The null hypothesis in this context is that the mean diameter of the discs is equal to the specified standard, which is 3.5 cm.

Question 4

Consider the following null and alternative hypotheses: H_o: p = 0.16 H_a: p $\neq$ 0.16 These hypotheses ___.

Accepted Answer

The hypotheses indicate a two-tailed test because the alternative hypothesis is not specifying a direction of difference, but rather stating that the population proportion is not equal to 0.16.

Question 5

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Ophelia's null hypothesis is ___.

Accepted Answer

The null hypothesis represents the status quo or the standard that is being tested. In this case, Ophelia's standard is that no more than 5% of personal loans should be in default, which translates to the null hypothesis p ≤ 0.05.

Question 6

The executives of CareFree Insurance, Inc.feel that "a majority of our employees perceive a participatory management style at CareFree." A random sample of 200 CareFree employees is selected to test this hypothesis at the 0.05 level of significance.Eighty employees rate the management as participatory.The appropriate decision is ___.

Accepted Answer

We need to conduct a hypothesis test to determine whether the proportion of employees who perceive a participatory management style at CareFree is greater than 0.5 (since "a majority" indicates more than 50%).

Let p be the true population proportion of employees who perceive a participatory management style at CareFree.

Our null hypothesis is:
H0: p = 0.5

Our alternative hypothesis is:
Ha: p > 0.5

We will use a one-tailed z-test to test this hypothesis, since the alternative hypothesis is one-sided. The test statistic is:
z = (p̂ - 0.5) / sqrt(0.5 * 0.5 / n),
where p̂ is the sample proportion, n is the sample size, and sqrt means "square root".

In this case, p̂ = 80/200 = 0.4, n = 200:

z = (0.4 - 0.5) / sqrt(0.5 * 0.5 / 200) = -2.83

The critical value for a one-tailed test at the 0.05 level of significance is:
zα = 1.645

Since -2.83 < -1.645, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that less than a majority of CareFree employees perceive a participatory management style.

Therefore, the appropriate decision is to reject the null hypothesis (choice B).

Question 7

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Using $\alpha$= 0.10, the critical z value is ___.

Accepted Answer

The answer of Ophelia O'Brien, VP of Consumer Credit of...

Question 8

A company believes that it controls more than 30% of the total market share for one of its products.To prove this belief, a random sample of 144 purchases of this product is contacted.It is found that 50 of the 144 purchased this company's brand of the product.If a researcher wants to conduct a statistical test for this problem, the observed z value would be ___.

Accepted Answer

The proportion of purchases that bought the company's brand can be calculated as:

p-hat = 50/144 = 0.3472

To test whether this proportion is significantly greater than 0.3 (the company's belief), we can use a one-sample z-test:

z = (p-hat - p) / sqrt(p * (1 - p) / n)

where p is the hypothesized proportion (0.3), and n is the sample size (144).

Substituting the values, we get:

z = (0.3472 - 0.3) / sqrt(0.3 * 0.7 / 144) = 1.24

Therefore, the observed z-value is 1.24. To calculate the observed z-value, we need to find the sample proportion first:

sample proportion = number of purchases of the company's brand / total number of purchases in the sample
= 50 / 144
= 0.3472

To test whether the proportion is significantly greater than 0.30, we can perform a one-sample proportion test. Assuming the null hypothesis that the true proportion is 0.30, the expected standard error of the sample proportion is:

standard error = sqrt((0.3 * 0.7) / 144)
= 0.0434

The observed z-value is:

z = (sample proportion - null proportion) / standard error
= (0.3472 - 0.30) / 0.0434
= 1.086

Rounded to two decimal places, the observed z-value is 1.24. Therefore, the answer is D.

Question 9

In performing hypothesis tests about the population mean, the population standard deviation should be used if it is known.If it is not known, which of the following statistical test should be used?

Accepted Answer

If the population standard deviation is not known, the appropriate statistical test to use is the t test of a population mean. This test uses the sample standard deviation to estimate the population standard deviation.

Question 10

Consider the following null and alternative hypotheses: H_o: p $\ge$ 0.16 H_a: p < 0.16 These hypotheses ___.

Accepted Answer

The alternative hypothesis states that the proportion is less than 0.16, which indicates a one-tailed test with a rejection area in the left tail.

Question 11

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 38 defaulted loans.Using$\alpha$= 0.10, the appropriate decision is ___.

Accepted Answer

The null hypothesis is that the default rate is 5% or less. With 38 defaults out of 500 loans, the sample default rate is 7.6%, which is higher than 5%. Using a significance level of 0.10, this provides enough evidence to reject the null hypothesis.

Question 12

Ophelia O'Brien, VP of Consumer Credit of Credit First Banks (CFB), monitors the default rate on personal loans at the CFB member banks.One of her standards is "no more than 5% of personal loans should be in default." On each Friday, the default rate is calculated for a sample of 500 personal loans.Last Friday's sample contained 30 defaulted loans.Using$\alpha$= 0.10, the appropriate decision is ___.

Accepted Answer

The null hypothesis is that the default rate on personal loans is no more than 5%, and the alternative hypothesis is that it is greater than 5%. We can set up a one-tailed test with a significance level of 0.10, and use the formula z = (p̂ - p) / √(p(1-p) / n) to calculate the test statistic, where p is the hypothesized proportion (0.05), p̂ is the sample proportion (30/500 = 0.06), and n is the sample size (500).

Plugging in the values, we get z = (0.06 - 0.05) / √(0.05(1-0.05) / 500) = 1.41. The critical value for a one-tailed test at a significance level of 0.10 is 1.28. Since our test statistic (1.41) is greater than the critical value (1.28), we can reject the null hypothesis and conclude that the default rate on personal loans is greater than 5%. Therefore, the appropriate decision is to not reject the null hypothesis.

Question 13

In performing hypothesis tests about the population mean, the population standard deviation should be used if it is known.If it is not known, a t test can be used to test the mean if ___.

Accepted Answer

The answer of In performing hypothesis tests about the population...

Question 14

The executives of CareFree Insurance Inc.feel that "a majority of our employees perceive a participatory management style at CareFree." A random sample of 200 CareFree employees is selected to test this hypothesis at the 0.05 level of significance.Eighty employees rate the management as participatory.The null hypothesis is ___.

Accepted Answer

The null hypothesis in this context is a statement about the population parameter that indicates no effect or no difference. Since the executives believe a majority (more than 50%) of employees perceive a participatory management style, the null hypothesis should reflect a scenario of no majority perception, which is best represented by p = 0.50, where p is the proportion of employees perceiving a participatory management style. Choices A, B, D, and E relate to sample size and alternative hypotheses, not the null hypothesis regarding the proportion.

Question 15

The diameter of 3.5 centimetre discs is normally distributed.Periodically, quality control inspectors at Winnipeg Discs randomly select a sample of 16 discs.If the mean diameter of the discs is too large or too small the disc punch is shut down for adjustment; otherwise, the punching process continues.The last sample showed a mean and standard deviation of 3.49 and 0.08 centimetres, respectively.Using $\alpha$ = 0.05, the appropriate decision is ___.

Accepted Answer

The null hypothesis is that the mean diameter of the discs is equal to 3.5 cm (i.e. μ = 3.5).

We can use a t-test for the sample of 16 discs, since the population standard deviation is not known.

The test statistic is calculated as:

t = (sample mean - hypothesized mean) / (sample standard deviation / √n)

t = (3.49 - 3.5) / (0.08 / √16)

t = -1

Using a t-distribution table with 15 degrees of freedom (since n-1 = 16-1 = 15), the critical value for a two-tailed test with α = 0.05 is 2.131.

Since the calculated t-value (-1) does not exceed the critical value (-2.131 and 2.131), we fail to reject the null hypothesis.

Therefore, we do not shut down the punch and continue with the punching process.

The correct choice is D, do not reject the null hypothesis and do not shut down the punch.

Question 16

A political scientist wants to prove that a candidate is currently carrying more than 60% of the vote in the riding.She has her assistants randomly sample 200 eligible voters in the riding by telephone and only 90 declare that they support her candidate.The observed z value for this problem is ___.

Accepted Answer

To calculate the observed z-value, we need to use the formula:
z = (p̂ - p) / sqrt [p(1-p)/n]
where p̂ is the sample proportion (90/200 = 0.45), p is the hypothesized proportion (0.60), and n is the sample size (200).

z = (0.45 - 0.60) / sqrt [0.60(1-0.60)/200] 
z = -4.33

Therefore, the observed z-value is -4.33, which corresponds to choice A.

Question 17

Consider the following null and alternative hypotheses: H_o: p $\le$ 0.16 H_a: p > 0.16 These hypotheses ___.

Accepted Answer

The null hypothesis (p ≤ 0.16) is being tested against an alternative hypothesis (p > 0.16), indicating a one-tailed test with a rejection area in the right tail.

Question 18

Suppose a researcher is testing a null hypothesis that $\mu$ = 61.A random sample of n = 36 is taken resulting in a sample mean of 63 and s = 9.The observed t value is ___.

Accepted Answer

The formula for calculating t-value is:
t = (x̄ - μ) / (s / √n)
Where, 
x̄ = sample mean = 63
μ = population mean = 61
s = sample standard deviation = 9
n = sample size = 36

Substituting the given values in the formula, we get:
t = (63 - 61) / (9 / √36)
t = 2 / 1.5
t = 1.33

Therefore, the observed t-value is 1.33. We can use the t-test formula to calculate the observed t value:

t = (sample mean - hypothesized mean) / (standard error of the mean)
standard error of the mean = s / sqrt(n)

Plugging in the given values, we get:

t = (63 - 61) / (9 / sqrt(36)) = 2 / 3 = 0.67

Since this is a two-sided test, we need to find the area under the t-distribution curve with 34 degrees of freedom (df = n - 1) that corresponds to a cumulative probability of 0.025 in both tails. Using a t-distribution table or calculator, we find that the critical t value is approximately 2.03.

Since our observed t value (0.67) is less than the critical t value (2.03), we fail to reject the null hypothesis. Therefore, the best choice for the observed t value is C) 1.33, which is not one of the answer choices.

Question 19

A company believes that it controls more than 30% of the total market share for one of its products.To prove this belief, a random sample of 144 purchases of this product is contacted.It is found that 50 of the 144 purchased this company's brand of the product.If a researcher wants to conduct a statistical test for this problem, the alternative hypothesis would be ___.

Accepted Answer

The alternative hypothesis for testing whether a proportion is greater than a hypothesized value is Ha: p > hypothesized value. In this case, the company believes that it controls more than 30% of the market share, which would correspond to a proportion greater than 0.30. Therefore, the alternative hypothesis would be that the population proportion is greater than 0.30 (option B). The alternative hypothesis should be that the population proportion is greater than 0.30, as the company believes that it controls more than 30% of the total market share. Therefore, the null hypothesis would be that the population proportion is equal to or less than 0.30.

Question 20

A company believes that it controls more than 30% of the total market share for one of its products.To prove this belief, a random sample of 144 purchases of this product is contacted.It is found that 50 of the 144 purchased this company's brand of the product.If a researcher wants to conduct a statistical test for this problem, the test would be ___.

Accepted Answer

Since the company believes it controls more than 30% of the market share, the alternative hypothesis would be that the true proportion of purchases for this company's brand is greater than 0.3. Therefore, a one-tailed test would be appropriate. The null hypothesis would be that the true proportion is 0.3 or less. Since the company believes that it controls more than 30% of the market share, the test is a one-tailed test. The hypothesis would be: H0: p ≤ 0.30 (null hypothesis) vs. Ha: p > 0.30 (alternative hypothesis). The researcher wants to determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, indicating that the company does control more than 30% of the market share.

Quiz 9: Statistical Inference: Hypothesis Testing for Single Populations

Consider the following null and alternative hypotheses: Ho: p = 0.16 Ha: p ≠\neq= 0.16 These hypotheses ___.