Q: Given a population proportion $$p = 0.45$$ and a sample size $$n = 200$$, what is the standard deviation of the sample proportion?

$$\$$\sqrt{\frac{0.45 \times 0.55}{200}} = 0.0351\$$

Question 1

Suppose the mean age of a population is $$\mu = 40$$ years with a standard deviation $$\sigma = 15$$ years. If a sample of $$n = 100 is $$taken, what is the standard error of the mean?

Accepted Answer

$$\$$\sigma_{\bar{x}} = \frac{15}{\sqrt{100}} = 1.5\$$

Question 2

A population has $$\mu = 90$$ and $$\sigma = 15. $$What is the probability that a sample mean from $$n = 100 is $$greater than $$93$$?

Accepted Answer

$$\P$$(\bar{x} \u003e 93) = 0.025\$$

Question 3

If the length of time to play a soccer match is normally distributed with $$\mu = 2$$ hours and $$\sigma = 0.5$$ hours, what is the probability that the sample mean for $$n = 50$$ games is less than $$1.9$$ hours?

Accepted Answer

$$\P$$(\bar{x} \u003c 1.9) = 0.0228\$$

Question 4

Given a population proportion $$p = 0.45$$ and a sample size $$n = 200$$, what is the standard deviation of the sample proportion?

Accepted Answer

$$\$$\sqrt{\frac{0.45 	imes 0.55}{200}} = 0.0351\$$

Question 5

A study finds that the average time to complete a tax form is $$\mu = 8.55$$ hours with $$\sigma = 2.25$$ hours. For a sample of $$n = 35$$, what is the probability that the sample mean is greater than $$11.28$$ hours?

Accepted Answer

$$\P$$(\bar{x} \u003e 11.28) = 0.0036\$$

Question 6

If the sample size increases, what happens to the standard error of the mean?

Accepted Answer

It decreases because $$\$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\$$

Question 7

A Gallup poll finds $$27\% of $$Americans believe moral values are getting worse. For a sample of $$n = 1000$$, what is the mean of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\$$\mu_{\hat{p}} = 0.27\$$

Question 8

Which of the following best describes the Central Limit Theorem as applied to sample means?

Accepted Answer

For large $$n$$, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Question 9

If $$p = 0.72$$ and $$n = 150$$, what is the probability that the sample proportion is less than $$0.70$$? (Use $$z = -0.54)$$

Accepted Answer

$$\P$$(\hat{p} \u003c 0.70) = 0.2946\$$

Question 10

A sample of $$n = 50$$ soccer matches has a mean time of $$2$$ hours and $$\sigma = 0.5$$ hours. What is the probability that the sample mean is between $$1.93$$ and $$2.12$$ hours?

Accepted Answer

$$\P(1.93$$ \u003c \bar{x} \u003c 2.12) = 0.8088\$$

Question 11

Which formula is used to calculate the standard deviation of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\$$\sqrt{\frac{p(1-p)}{n}}\$$

Question 12

If a population is not normally distributed, under what condition is the sampling distribution of the sample mean approximately normal?

Accepted Answer

When the sample size $$n is $$large (typically $$n \geq 30)$$

Question 13

A sample of $$n = 200 is $$drawn from a population with $$p = 0.45. $$What is the probability that the sample proportion is greater than $$0.52$$? (Use $$z = 2.00)$$

Accepted Answer

$$\P$$(\hat{p} \u003e 0.52) = 0.0228\$$

Question 14

If the mean time to complete a tax form is $$8.55$$ hours with a standard deviation of $$2.25$$ hours, what is the $$95th $$percentile for the sample mean for $$n = 35$$?

Accepted Answer

$$\X = 8.55 + 1.645$$ 	imes \frac{2.25}{\sqrt{35}} = 9.14\$$

Question 15

A population has $$\mu = 35$$ and $$\sigma = 9. $$For a sample of $$n = 100$$, what is the probability that the sample mean is more than $$32$$?

Accepted Answer

$$\P$$(\bar{x} \u003e 32) = 0.99865\$$

Sampling Distributions and the Normal Probability Distribution: Study Notes

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