Question 1

i. For a one-tailed test using the 0.05 level of significance, the critical value for the z test is 1.645, but for t it is 1.96.
Ii) For a one-tailed test using the 0.01 level of significance, the critical value for the z-test is 1.645, but for t it is 1.96.
Iii) For a two-tailed test using the 0.05 level of significance the critical value for the z-test is 1.96 and it is the same for the t-test.

Accepted Answer

The answer of i. For a one-tailed test using the...

Question 2

It is claimed that in a bushel of peaches less than ten percent 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?

Accepted Answer

The alternate hypothesis for a one-sided test is that the proportion of defective peaches is less than 10%, or p < 0.10.

Question 3

Records on a fleet of trucks reveal that the average life of a set of spark plugs is normally distributed with a mean of 35,600 kilometres. A manufacturer of spark plugs claims that its plugs have an average life in excess of 35,600 kilometres. The fleet owner purchased 18 sets and found that the sample average life was 37,700 kilometres, the sample standard deviation was 2415 kilometres and the computed t = 3.677.

Accepted Answer

The computed t-value of 3.677 is likely greater than the critical t-value at the 0.05 significance level for a one-tailed test with 17 degrees of freedom, indicating sufficient evidence to support the manufacturer's claim.

Question 4

A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in a sample mean of
$220 and a standard deviation of $12. The t-test is to be conducted at the 5% level of significance. The t-value is calculated to be 5.77. At the 0.01 level of significance, what is your decision?

Accepted Answer

With a t-value of 5.77 and a sample size of 12, we have a t-test statistic of 3.055, which is greater than the critical t-value of 2.718 for a one-tailed test with 11 degrees of freedom at the 0.01 level of significance. This means we reject the null hypothesis and conclude that the average monthly bill exceeds $200. The z-test is not applicable here because we do not know the population standard deviation.

Question 5

What are the critical z-values for a two-tailed hypothesis test if α = 0.01?

Accepted Answer

For a two-tailed test with α = 0.01, the critical z-values are ±2.58, as this corresponds to the 0.5% in each tail of the standard normal distribution.

Question 6

The mean gross annual incomes of certified tack welders are normally distributed with the mean of $50,000 and a standard deviation of $4,000. The ship building association wishes to find out whether their tack welders earn more or less than $50,000 annually. The alternate hypothesis is that the mean is not $50,000.
Which of the following is the alternate hypothesis?

Accepted Answer

The alternate hypothesis is that the mean is not $50,000. The symbol for the population mean is μ, so the correct answer is B: μ ≠ $50,000.

Question 7

A manufacturer wants to increase the absorption capacity of a sponge. Based on past data, the average sponge could absorb 103.5ml. After the redesign, the absorption amounts of a sample of sponges were (in millilitres): 121.3, 109.2, 97.6, 103.5, 112.4, 115.3, 106.5, 112.4, 118.3, and 115.3. What is the decision rule at the 0.01 level of significance to test if the new design increased the absorption amount of the sponge?

Accepted Answer

The answer of A manufacturer wants to increase the absorption...

Question 8

What is the critical value for a one-tailed hypothesis test in which a null hypothesis is tested at the 5% level of significance based on a sample size of 25?

Accepted Answer

The answer of What is the critical value for a...

Question 9

What are the two rejection areas in using a two-tailed test and the 0.01 level of significance when n is large?

Accepted Answer

In a two-tailed test with a 0.01 level of significance, the critical z-values are approximately ±2.58, which define the rejection regions.

Question 10

A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in a sample mean of
$220 and a standard deviation of $12. The t-test is to be conducted at the 5% level of significance.

Accepted Answer

The null hypothesis should state that the average monthly bill is $200 or less (μ ≤ 200), and the alternate hypothesis should state that the average monthly bill exceeds $200 (μ > 200), which aligns with the restaurant's concern.

Question 11

What is the critical value for a one-tailed hypothesis test in which a null hypothesis is tested at the 5% level of significance based on a sample size of 25?

Accepted Answer

The answer of What is the critical value for a...

Question 12

From past records it is known that the average life of a battery used in a digital clock is 305 days. The battery life is normally distributed. The battery was recently modified to last longer. A sample of 20 of the modified batteries was tested. It was discovered that the mean life was 311 days and the sample standard deviation was 12 days. We want to test at the 0.05 level of significance whether the modification increases the life of the battery. What is our decision rule?

Accepted Answer

The answer of From past records it is known that...

Question 13

A random sample of size 15 is selected from a normal population. The population standard deviation is unknown. Assume that a two-tailed test at the 0.10 significance level is to be used. For what value of t will the null hypothesis be rejected?

Accepted Answer

For a two-tailed test with a sample size of 15 (degrees of freedom = 14) and a significance level of 0.10, the critical t-values are approximately -1.761 and 1.761.

Question 14

i. If the null hypothesis is μ ≥ 200 and the alternate hypothesis states that μ is less than 200, then, a two-tail test is being conducted.
Ii) For a one-tailed test of hypothesis, the area of rejection is only in one tail of the curve.
Iii) As the sample size increases, the curve of the t-distribution approaches the standard normal distribution

Accepted Answer

Statement (i) is incorrect because the hypotheses describe a one-tailed test. Statement (ii) is correct as a one-tailed test has the rejection area in one tail. Statement (iii) is correct because as sample size increases, the t-distribution approaches the normal distribution.

Question 15

A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in a sample mean of
$220 and a standard deviation of $12. The t-test is to be conducted at the 5% level of significance.

Accepted Answer

The answer of A restaurant that bills its house account...

Question 16

A machine is set to fill the small size packages of Smarties candies with 56 candies per bag. A sample revealed: 3 bags of 56, 2 bags of 57, 1 bag of 55, and 2 bags of 58. How many degrees of freedom are there?

Accepted Answer

The degrees of freedom for a sample is calculated as the number of observations minus one. Here, there are 8 bags, so the degrees of freedom is 8 - 1 = 7.

Question 17

A manufacturer wants to increase the shelf life of a line of cake mixes. Past records indicate that the average shelf life of the mix is 216 days. After a revised mix has been developed, a sample of nine boxes of cake mix gave these shelf lives (in days): 215, 217, 218, 219, 216, 217, 217, 218 and 218. At the 0.025 level, has the shelf life of the cake mix increased?

Accepted Answer

The computed t-value is greater than the critical value, indicating that the shelf life of the cake mix has increased.

Question 18

A nationwide survey of college students was conducted and found that students spend two hours per class hour studying. A professor at your school wants to determine whether the time students spend at your school is significantly different from the two hours. A random sample of fifteen statistics students is carried out and the findings indicate an average of 1.75 hours with a standard deviation of 0.24 hours.

Accepted Answer

The null hypothesis (H0) is the statement that there is no difference between the population mean and the hypothesized value. The alternative hypothesis (H1) is the statement that there is a difference between the population mean and the hypothesized value. In this case, the hypothesized value is 2 hours. The null hypothesis is that the population mean is equal to 2 hours, and the alternative hypothesis is that the population mean is not equal to 2 hours.

Question 19

A random sample of size 15 is selected from a normal population. The population standard deviation is unknown. Assume that a two-tailed test at the 0.10 significance level is to be used. For what value of t will the null hypothesis be rejected?

Accepted Answer

The answer of A random sample of size 15 is...

Question 20

If the alternative hypothesis states that μ > 6,700, what is the rejection region for the hypothesis test?

Accepted Answer

The alternative hypothesis μ > 6,700 indicates a one-tailed test in the upper tail, where we are looking for evidence that the mean is greater than 6,700.

Quiz 9: One-Sample Tests of a Hypothesis

It is claimed that in a bushel of peaches less than ten percent 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?

What are the critical z-values for a two-tailed hypothesis test if α = 0.01?

What is the critical value for a one-tailed hypothesis test in which a null hypothesis is tested at the 5% level of significance based on a sample size of 25?

What are the two rejection areas in using a two-tailed test and the 0.01 level of significance when n is large?

A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in a sample mean of $220 and a standard deviation of $12. The t-test is to be conducted at the 5% level of significance.

What is the critical value for a one-tailed hypothesis test in which a null hypothesis is tested at the 5% level of significance based on a sample size of 25?

A random sample of size 15 is selected from a normal population. The population standard deviation is unknown. Assume that a two-tailed test at the 0.10 significance level is to be used. For what value of t will the null hypothesis be rejected?

A restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve accounts is selected, resulting in a sample mean of $220 and a standard deviation of $12. The t-test is to be conducted at the 5% level of significance.

A machine is set to fill the small size packages of Smarties candies with 56 candies per bag. A sample revealed: 3 bags of 56, 2 bags of 57, 1 bag of 55, and 2 bags of 58. How many degrees of freedom are there?

A random sample of size 15 is selected from a normal population. The population standard deviation is unknown. Assume that a two-tailed test at the 0.10 significance level is to be used. For what value of t will the null hypothesis be rejected?

If the alternative hypothesis states that μ > 6,700, what is the rejection region for the hypothesis test?