Question

1. Which of the following distributions is used when estimating the population mean from a normal population with unknown variance?

A

the t distribution with n + 1 degrees of freedom

B

the t distribution with n degrees of freedom

C

the t distribution with n - 1 degrees of freedom

D

the t distribution with 2n degrees of freedom

2. If a sample has 20 observations and a 90% confidence estimate for μ is needed, the appropriate t-score is:

A

2.120

B

1.746

C

2.131

D

1.729

3. If a researcher fails to reject the null hypothesis, then the alternative hypothesis should be accepted.

A

True

B

False

C

Not sure

4. The significance level and the probability of a Type I error are equivalent.

A

True

B

False

C

Not sure

5. Increasing the significance level of a test increases the probability of a Type II error.

A

True

B

False

C

Not sure

6. Explain the relationship between Type I error and Type II error in a hypothesis test.

Answer 1

Answer: 1. Which of the following distributions is used when estimating the population mean from a normal population with unknown variance?

Solution:

t distribution is used for sample size (n < 30) and unknown population variance.

Therefore,

the t distribution with n - 1 degrees of freedom is used when estimating the population mean from a normal population with unknown variance.

The option C is correct answer.

Answer 2:. If a sample has 20 observations and a 90% confidence estimate for μ is needed, the appropriate t-score is:

Solution: At 90% confidence interval, α = 0.10

df = n-1 = 20 - 1 = 19

t critical = t(α/2,df) = t(0.05,19)

t critical = 1.7291 (from t table).

Therefore, t critical = 1.729.

The option D is correct answer.

Answer 3: If a researcher fails to reject the null hypothesis, then the alternative hypothesis should be accepted.

This statement is False.

If a researcher fails to reject the null hypothesis, then the alternative hypothesis should be rejected or null hypothesis should be accepted.

Answer 4. The significance level and the probability of a Type I error are equivalent.

This statement is​​​​​True.

The probability of committing a type 1 error denoted by α is referred to as the level of significance of the statistical test.

Answer 5: Increasing the significance level of a test increases the probability of a Type II error.

This statement is False.

For a given level of α increasing the sample size decreases type II error. or we can say that probability of type II error decreases when sample size increases.

Answer 6: Explain the relationship between Type I error and Type II error in a hypothesis test.

Solution:

The relationship between Type I error and Type II error in a hypothesis test:

Statistical Decision	Ho True	Ho False
Do not reject Ho	correct decision, Confidence =(1 - α)	Type II error, P(Type II error) =β
Reject Ho	Type I error, P(Type I error) = α	correct decision, power = (1-β)

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