Question 1

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 critical value for α = 0.025?

Accepted Answer

The answer of It is claimed that in a bushel...

Question 2

What is the level of significance?

Accepted Answer

The level of significance is the probability of making a Type I error, which is rejecting a true null hypothesis.

Question 3

i. Two examples of a hypothesis are: 
1) mean monthly income from all sources for senior citizens is $841 and 
2) twenty percent of juvenile offenders ultimately are caught and sentenced to prison. 
ii. Hypothesis testing is a procedure based on sample evidence and probability theory to decide whether the hypothesis is a reasonable statement.
Iii) We call a statement about the value of a population parameter a hypothesis.

Accepted Answer

All three statements are correct and accurately describe hypotheses and hypothesis testing.

Question 4

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 sample proportion?

Accepted Answer

The sample proportion is calculated by dividing the number of defective peaches by the total number of peaches in the sample: 50/400 = 0.125.

Question 5

What is the probability of making a Type II error if the null hypothesis is actually true?

Accepted Answer

A Type II error occurs when the null hypothesis is false but is incorrectly accepted. If the null hypothesis is true, the probability of making a Type II error is 0.

Question 6

i. The level of significance is the risk we assume of rejecting the null hypothesis when it is actually true.
Ii) There is only one level of significance that is applied to all studies involving sampling.
Iii) The researcher must decide on the level of significance before formulating a decision rule and collecting sample data.

Accepted Answer

The answer of i. The level of significance is the...

Question 7

What is a Type II error?

Accepted Answer

Accepting a false null hypothesis is a Type II error.

Question 8

i. If the critical values of the test statistic z are ±1.96, they are the dividing points between the areas of rejection and non-rejection.
Ii) The probability of a Type I error is also referred to as alpha.
Iii) A Type I error is the probability of rejecting a true null hypothesis.

Accepted Answer

The critical values ±1.96 are indeed the dividing points for rejection and non-rejection in a standard normal distribution. The probability of a Type I error is denoted by alpha, and a Type I error occurs when a true null hypothesis is incorrectly rejected.

Question 9

i. Two types of possible errors always exist when testing hypotheses-a Type I error, in which the null hypothesis is rejected when it should not have been rejected, and a Type II error in which the null hypothesis is not rejected when it should have been rejected.
Ii) A test statistic is a value determined from sample information collected to test the null hypothesis. 
iii. The region or area of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote.

Accepted Answer

All three statements are correct and accurately describe different aspects of hypothesis testing. Statement (i) defines the two types of errors that can occur, statement (ii) defines what a test statistic is, and statement (iii) defines the region of rejection.

Question 10

i. The probability of a Type I error is also referred to as alpha. 
ii. A Type I error is the probability of accepting a true null hypothesis. 
iii. A Type I error is the probability of rejecting a true null hypothesis.

Accepted Answer

Statement (i) is correct as alpha represents the probability of a Type I error. Statement (iii) is correct because a Type I error occurs when a true null hypothesis is rejected. Statement (ii) is incorrect because it describes a Type II error.

Question 11

i. An alternate hypothesis is a statement about a population parameter that is accepted when the null hypothesis is rejected.
Ii) The level of significance is the risk we assume of rejecting the null hypothesis when it is actually true.
Iii) The researcher must decide on the level of significance before formulating a decision rule and collecting sample data.

Accepted Answer

All three statements are correct. The alternate hypothesis is accepted when the null hypothesis is rejected. The level of significance is indeed the risk of rejecting a true null hypothesis, and it should be decided before data collection.

Question 12

i. The fifth and final step in testing a hypothesis is taking a sample and, based on the decision rule, deciding if the null hypothesis should be rejected. 
ii. When the null hypothesis is not rejected, the conclusion is that our sample data does not allow us to reject the null hypothesis.
Iii) The level of significance is selected after setting up a decision rule and sampling the population.

Accepted Answer

Statement (i) and (ii) are correct. Statement (iii) is incorrect because the level of significance is selected before setting up a decision rule and sampling the population.

Question 13

i. To set up a decision rule, the sampling distribution is divided into two regions-a region of non- rejection and a region where the null hypothesis is rejected.
Ii) A test statistic is a value determined from sample information collected to test the null hypothesis. 
iii. If the null hypothesis is true and the researchers do not reject it, then a correct decision has been made.

Accepted Answer

All three statements are correct. Statement (i) describes the basic concept of hypothesis testing, statement (ii) defines what a test statistic is, and statement (iii) describes a correct decision in hypothesis testing when the null hypothesis is true and not rejected.

Question 14

i. An alternate hypothesis is a statement about a population parameter that is accepted when the null hypothesis is rejected. 
ii. The level of significance is the risk we assume of rejecting the null hypothesis when it is actually true.
Iii) There is only one level of significance that is applied to all studies involving sampling.

Accepted Answer

Statement (i) and (ii) are correct. The alternate hypothesis is accepted when the null hypothesis is rejected, and the level of significance is indeed the risk of rejecting a true null hypothesis. Statement (iii) is incorrect because different studies can have different levels of significance.

Question 15

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.
Given the following megastat printout, what conclusions can be made?

Accepted Answer

The answer of The mean gross annual incomes of certified...

Question 16

i. If the critical values of the test statistic z are ±1.96, they are the dividing points between the areas of rejection and non-rejection.
Ii) The probability of a Type I error is also referred to as alpha.
Iii) A Type I error is the probability of accepting a true null hypothesis.

Accepted Answer

The critical values ±1.96 are indeed the dividing points for rejection and non-rejection areas in a standard normal distribution. The probability of a Type I error is referred to as alpha. However, a Type I error is the probability of rejecting a true null hypothesis, not accepting it.

Question 17

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 z-statistic?

Accepted Answer

The z-statistic is calculated using the formula: z = (p̂ - p) / √(p(1-p)/n), where p̂ is the sample proportion (50/400 = 0.125), p is the population proportion (0.10), and n is the sample size (400). Plugging in these values gives z = (0.125 - 0.10) / √(0.10*0.90/400) = 1.67.

Question 18

i. Two types of possible errors always exist when testing hypotheses-a Type I error, in which the null hypothesis is rejected when it should not have been rejected, and a Type II error in which the null hypothesis is not rejected when it should have been rejected.
Ii) A test statistic is a value determined from sample information collected to test the null hypothesis. 
iii. If we do not reject the null hypothesis based on sample evidence, we have proven beyond doubt that the null hypothesis is true.

Accepted Answer

The answer of i. Two types of possible errors always...

Question 19

i. If we do not reject the null hypothesis based on sample evidence, we have proven beyond doubt that the null hypothesis is true.
Ii) A test statistic is a value determined from sample information collected to test the null hypothesis. 
iii. The region or area of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote.

Accepted Answer

The answer of i. If we do not reject the...

Question 20

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.
If α = 0.025, what will be the decision?

Accepted Answer

The answer of It is claimed that in a bushel...

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 critical value for α = 0.025?

What is the level of significance?

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 sample proportion?

What is the probability of making a Type II error if the null hypothesis is actually true?

What is a Type II error?

i. If the critical values of the test statistic z are ±1.96, they are the dividing points between the areas of rejection and non-rejection. Ii) The probability of a Type I error is also referred to as alpha. Iii) A Type I error is the probability of rejecting a true null hypothesis.

i. The probability of a Type I error is also referred to as alpha. ii. A Type I error is the probability of accepting a true null hypothesis. iii. A Type I error is the probability of rejecting a true null hypothesis.

i. If the critical values of the test statistic z are ±1.96, they are the dividing points between the areas of rejection and non-rejection. Ii) The probability of a Type I error is also referred to as alpha. Iii) A Type I error is the probability of accepting a true null 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 z-statistic?

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. If α = 0.025, what will be the decision?