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AIM 5: Explain the degree of freedom

1、With 60 observations, what is the appropriate number of degrees of freedom to use when carrying out a statistical test on the mean of a population?

A) 61.

B) 60.

C) 58.

D) 59.

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The correct answer is D

When performing a statistical test on the mean of a population based on a sample of size n, the number of degrees of freedom is n – 1 since once the mean is estimated from a sample there are only n – 1 observations that are free to vary. In this case the appropriate number of degrees of freedom to use is 60 – 1 = 59.

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2、Which one of the following distributions is described entirely by the degrees of freedom?

A) Student’s t-distribution.

B) Normal distribution.

C) Binomial distribution.

D) Lognormal distribution.

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2、Which one of the following statements about the t-distribution is most accurate?

A) The t-distribution is the appropriate distribution to use when constructing confidence intervals based on large samples.

B) The t-distribution approaches the standard normal distribution as the number of degrees of freedom becomes large.

C) Compared to the normal distribution, the t-distribution is more peaked with more area under the tails.

D) Compared to the normal distribution, the t-distribution is flatter with less area under the tails.

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The correct answer is B

As the number of degrees of freedom grows, the t-distribution approaches the shape of the standard normal distribution. Compared to the normal distribution, the t-distribution is less peaked with more area under the tails. When choosing a distribution, three factors must be considered: sample size, whether population variance is known, and if the distribution is normal.

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3、When is the t-distribution the appropriate distribution to use? The t-distribution is the appropriate distribution to use when constructing confidence intervals based on:

A) small samples from populations with known variance that are at least approximately normal.

B) large samples from populations with known variance that are nonnormal.

C) large samples from populations with known variance that are at least approximately normal.

D) small samples from populations with unknown variance that are at least approximately normal

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The correct answer is D

The t-distribution is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance that are either normal or approximately normal.

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The correct answer is D

As n gets larger, the variance of the distribution of sample means is reduced, and the distribution of sample means approximates a normal distribution.

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3、To apply the central limit theorem to the sampling distribution of the sample mean, the sample is usually considered to be large if n is greater than:

A) 15.

B) 20.

C) 30.

D) 25.

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The correct answer is C

A sample size is considered sufficiently large if it is larger than or equal to 30.