Q: In a population where $$p = 0.52$$ and $$n = 300$$, what is the standard deviation of the sampling distribution of the sample proportion?

$$\sigma_{\hat{p}} = \sqrt{\frac{0.52 \times 0.48}{300}} \approx 0.0288$$

Question 1

Which of the following statements best describes the Empirical Rule (68-95-99.7 Rule) for a normal distribution?

Accepted Answer

About 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.

Question 2

Suppose the cholesterol levels in a population are normally distributed with mean $$\mu = 200 mg/100ml $$and standard deviation $$\sigma = 20 mg/100ml. $$What is the Z-score for a cholesterol value of 150 mg/100ml?

Accepted Answer

$$z = \frac{150 - 200}{20} = -2.5$$

Question 3

A sample of 25 students has SAT scores that are normally distributed with mean $$\mu = 1215$$ and standard deviation $$\sigma = 123. $$Ben scored 1350. What is Ben's Z-score?

Accepted Answer

$$z = \frac{1350 - 1215}{123} \approx 1.10$$

Question 4

If a value is more than 2 standard deviations away from the mean in a normal distribution, how is it typically classified?

Accepted Answer

It is considered an unusual value.

Question 5

Given a normal distribution $$N(3500, 500)$$ for infant birth weights, what percentage of birth weights are greater than 3500g?

Accepted Answer

50%

Question 6

Which formula is used to standardize a value $$X$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$z = \frac{X - \mu}{\sigma}$$

Question 7

A population has $$\mu = 90$$ and $$\sigma = 12. $$What is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{X}} = \frac{12}{\sqrt{25}} = 2.4$$

Question 8

Which statement best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean will be approximately normal if the sample size is large, regardless of the population's distribution.

Question 9

In a population where $$p = 0.52$$ and $$n = 300$$, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.52 	imes 0.48}{300}} \approx 0.0288$$

Question 10

A researcher wants to find the probability that the difference between two sample means is greater than 8, given $$\mu_A = 188$$, $$\mu_B = 194$$, $$\sigma_A = 37.8$$, $$\sigma_B = 38.5$$, $$n_A = 60$$, $$n_B = 60. $$What is the Z-score for this difference?

Accepted Answer

$$z = \frac{8 - (188 - 194)}{\sqrt{\frac{37.8^2$}{60} + \frac{38.5^2$}{60}}}$$

Question 11

Which condition must be met for the sampling distribution of the sample proportion to be approximately normal?

Accepted Answer

Both $$np$$ and $$n(1-p)$$ must be greater than 5.

Question 12

If the area under the standard normal curve to the left of $$z = 1.25 is 0.8944$$, what is the probability that $$z \u003e 1.25$$?

Accepted Answer

$$P(z \u003e 1.25) = 1 - 0.8944 = 0.1056$$

Question 13

A sample of 200 individuals from Population A (proportion $$p_1 = 0.36$) and 210 from Population B (p_2$ = 0.28) is $$taken. What is the standard deviation of the sampling distribution of the difference in proportions?

Accepted Answer

$$\sigma_{p_1$ - p_2$} = \sqrt{\frac{0.36 	imes 0.64}{200} + \frac{0.28 	imes 0.72}{210}}$$

Question 14

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is skewed to the right.

Question 15

If you want to find the value $$x$$ such that 85% of a normal population has values less than $$x$$, which steps should you follow?

Accepted Answer

Find the z-score corresponding to 0.85 in the Z-table, then use $$x = z \sigma + \mu.$$

Question 16

Which graphical method is commonly used to assess normality in a dataset using statistical software?

Accepted Answer

Q-Q plot

Question 17

If the sample size increases, what happens to the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

It decreases.

Module 3: Normal Distributions & Sampling Distributions – Study Notes

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