Question 1

Human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F. If a sample of 19 people is randomly selected, what is the probability that their mean body temperature will be less than 98.50°F? Use the Central Limit Theorem.

Accepted Answer

$$0.9826$$

Question 2

Which formula should be used to calculate the standard error of the mean for a sample of size $$n$$ from a population with standard deviation $$\sigma$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 3

Suppose the mean body temperature is $$\mu = 98.20$$°F and the standard deviation is $$\sigma = 0.62$$°F. For a sample of $$n = 19$$, what is the $$z-$$score for a sample mean of $$98.50$$°F?

Accepted Answer

$$z = \frac{98.50 - 98.20}{0.62 / \sqrt{19}}$$

Question 4

A hospital wants to know if the average body temperature of its patients is higher than the national average of $$98.20$$°F. They collect a sample of $$n = 19$$ patients and find a mean of $$98.50$$°F. Which statistical test is most appropriate?

Accepted Answer

A one-sample $$z-$$test for the mean

Question 5

If the probability that the sample mean is less than $$98.50$$°F is $$0.9826$$, what does this imply about the sample mean in relation to the population mean?

Accepted Answer

The sample mean is very likely to be less than $$98.50$$°F

Question 6

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The distribution of sample means approaches a normal distribution as sample size increases, regardless of the population's distribution

Question 7

Given a population with mean $$\mu$$ and standard deviation $$\sigma$$, what is the mean and standard deviation of the sampling distribution of the sample mean for samples of size $$n$$?

Accepted Answer

Mean: $$\mu$$, Standard deviation: $$\frac{\sigma}{\sqrt{n}}$$

Question 8

A researcher claims that the average body temperature is higher than $$98.20$$°F. If the sample mean is $$98.50$$°F with $$n = 19$$ and $$\sigma = 0.62$$°F, what is the $$p-$$value for the test statistic $$z = 2.13$$?

Accepted Answer

$$p = 0.0166$$

Question 9

If the sample mean is $$98.50$$°F, population mean is $$98.20$$°F, $$\sigma = 0.62$$°F, and $$n = 19$$, what is the probability that the sample mean is less than $$98.50$$°F?

Accepted Answer

$$0.9826$$

Question 10

Which of the following is NOT an assumption required for using the Central Limit Theorem?

Accepted Answer

The sample mean must equal the population mean

Question 11

A medical researcher wants to estimate the probability that the mean body temperature of a random sample of 19 adults exceeds $$98.50$$°F. Which calculation should be performed first?

Accepted Answer

Calculate the $$z-$$score for $$98.50$$°F

Question 12

If the standard deviation of body temperature is $$0.62$$°F, what is the standard error of the mean for a sample of $$n = 19$$?

Accepted Answer

$$0.142$$°F

Question 13

Which of the following statements is TRUE regarding the distribution of sample means for body temperature?

Accepted Answer

It will be approximately normal if the sample size is large enough

Question 14

A sample of 19 people has a mean body temperature of $$98.50$$°F. If the population mean is $$98.20$$°F and $$\sigma = 0.62$$°F, what is the correct formula to find the probability that the sample mean is less than $$98.50$$°F?

Accepted Answer

$$P(\bar{x} \u003c 98.50) = P\left(z \u003c \frac{98.50 - 98.20}{0.62 / \sqrt{19}}\right)$$

Question 15

If the population of body temperatures is normally distributed, what is the shape of the sampling distribution of the sample mean for any sample size?

Accepted Answer

Normal

Question 16

A doctor wants to know the likelihood that a randomly selected group of 19 patients will have an average body temperature above $$98.50$$°F. What is the correct approach?

Accepted Answer

Calculate the $$z-$$score and use the standard normal table to find $$P(\bar{x} \u003e 98.50)$$

Normal Distribution and Sampling Distributions: Probability of Sample Mean

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