Question 1

If the height of 20-year-old American women is normally distributed with a mean of 64 inches and standard deviation of 3 inches, what is the standard deviation of the sampling distribution for the average height of a random sample of 36 women?

Accepted Answer

$$\frac{3}{\sqrt{36}} = 0.5$$

Question 2

A physician takes a random sample of 16 adult US women and finds their average diastolic blood pressure. If the population mean is 80 mm Hg and the standard deviation is 12 mm Hg, what is the standard error of the mean for this sample?

Accepted Answer

$$\frac{12}{\sqrt{16}} = 3$$

Question 3

A restaurant’s records show that the cost of customers’ dinners has a skewed distribution with a mean of $$24 and a standard deviation of 15. $$What is the probability that the next 100 customers will have an average dinner cost of at least $30?

Accepted Answer

$$1 - pnorm(30, mean=24, sd=1.5)$$

Question 4

Earlier this month, 65 Californians voted 'yes' on Measure H. A state task team wants to estimate the probability that in a sample of 50 Californians, more than 65% will vote 'yes' on Measure H. Which formula should be used?

Accepted Answer

$$1 - pnorm(0.65, mean=0.54, sd=0.1)$$

Question 5

For all adult women in the US, the diastolic blood pressure is normally distributed with mean 80 mm Hg and standard deviation 12 mm Hg. A physician takes a random sample of 16 women and finds their average diastolic blood pressure is 90 mm Hg. What is the z-score for this sample mean?

Accepted Answer

$$z = \frac{90 - 80}{12 / \sqrt{16}} = \frac{10}{3} = 3.33$$

Question 6

The number of miles driven per month by a household follows a probability distribution with mean 820 and standard deviation 80. What is the probability that in one of the next 9 months, the household has driven more than 850 miles?

Accepted Answer

$$1 - pnorm(850, mean=820, sd=80)$$

Question 7

Compute the probability that the average number of miles driven in three 9-month periods is less than 825, given mean $$\mu = 820$$ and $$\sigma = 80.$$

Accepted Answer

$$pnorm(825, mean=820, sd=46)$$

Question 8

The price of any single selected house in a community follows a probability distribution with mean $$190,000$$ USD and standard deviation $$25,000$$ USD. A real estate agent has a list of 4 unsold houses from that community. What is the mean and standard deviation of the average price of all the houses in that list?

Accepted Answer

Mean = $$190,000$$ USD, standard deviation = $$12,500$$ USD

Question 9

A researcher finds that TAMU students spend 12.4 hours per week browsing the internet, with a standard deviation of 2.8 hours. If 20 students are randomly selected, what is the standard error of the mean for the sample?

Accepted Answer

$$\frac{2.8}{\sqrt{20}} = 0.626$$

Question 10

For a grocery store in College Station, is it true that 16% of ALL customers are from outside the city limits? If Bryan-College Station has randomly selected 35 customers and 12 are from outside the city limits, what is the sample proportion?

Accepted Answer

$$\frac{12}{35} = 0.343$$

Question 11

Suppose you want to know whether the average income of any household within a rural community is at least $$USD 3000$$ and the standard deviation is $$USD 500. A $$study plan takes 50 households in that community and their income in the preceding month. What is the standard error of the mean for this sample?

Accepted Answer

$$\frac{500}{\sqrt{50}} = 70.71$$

Question 12

A DVD rental store wants to know what proportion of its customers are under age 21. A simple random sample of 60 customers is taken, with 40 out of 60 under age 21. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{40}{60} = 0.667$$

Question 13

A researcher claims that the average amount of time spent by 36 students browsing the internet last week is more than 14 hours. The population mean is 12.4 hours and standard deviation is 2.8 hours. What is the z-score for the sample mean of 14 hours?

Accepted Answer

$$z = \frac{14 - 12.4}{2.8 / \sqrt{36}} = \frac{1.6}{0.467} = 3.43$$

Question 14

Suppose that 100 items are drawn from a population of manufactured products and the weight, in oz, is recorded. Prior experience has shown that the weight has a right-skewed probability distribution with mean 40 oz and standard deviation 9 oz. Find the value of $$P(\bar{X} \u003e 50).$$

Accepted Answer

$$1 - pnorm(50, mean=40, sd=0.9)$$

Question 15

A sample of 25 houses in a community has a mean price of $$190,000$$ USD and a standard deviation of $$25,000$$ USD. What is the standard error of the mean price?

Accepted Answer

$$\frac{25,000}{\sqrt{25}} = 5,000$$

Question 16

A researcher wants to estimate the probability that the average amount of time spent by 36 students browsing the internet last week is more than 14 hours. The population mean is 12.4 hours and standard deviation is 2.8 hours. Which formula should be used?

Accepted Answer

$$1 - pnorm(14, mean=12.4, sd=0.467)$$

Question 17

A sample of 60 customers at a DVD rental store shows that 44 are under age 21. What is the sample proportion $$\hat{p}$$ and its standard error?

Accepted Answer

$$\hat{p} = \frac{44}{60} = 0.733$$, $$SE = \sqrt{\frac{0.733 	imes 0.267}{60}} = 0.057$$

Question 18

A researcher wants to know if the sample mean of a skewed distribution can be approximated by a normal distribution. Which condition must be met for the Central Limit Theorem to apply?

Accepted Answer

The sample size $$n$$ must be at least 30.

Sampling Distributions, Normal Approximation, and Probability in Statistics

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