Q: A sample of 1000 Americans shows that 88% support solar energy. What is the standard error of the sample proportion?

$$SE_{\hat{p}} = \sqrt{\frac{0.88 \times 0.12}{1000}} = 0.01$$

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A single value calculated from sample data to estimate a population parameter.

Question 2

Suppose you survey 200 randomly selected adults and find that 150 support the expansion of solar energy. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{150}{200} = 0.75$$

Question 3

Which condition must be met for the Central Limit Theorem (CLT) to apply to sample means from any distribution?

Accepted Answer

The sample size $$n$$ must be greater than or equal to 30.

Question 4

Given a population mean $$\mu = 78$$ and standard deviation $$\sigma = 13.5$$, what is the standard error of the mean for a sample size of $$n = 10$$?

Accepted Answer

$$SE = \frac{13.5}{\sqrt{10}} = 4.27$$

Question 5

If the sample proportion $$\hat{p} is $$normally distributed with mean $$p$$ and standard error $$SE_{\hat{p}}$$, which formula gives the probability that $$\hat{p} is $$less than a value $$x$$?

Accepted Answer

$$P(\hat{p} \u003c x) = P\left(z \u003c \frac{x - p}{SE_{\hat{p}}}\right)$$

Question 6

A sample of 1000 Americans shows that 88% support solar energy. What is the standard error of the sample proportion?

Accepted Answer

$$SE_{\hat{p}} = \sqrt{\frac{0.88 	imes 0.12}{1000}} = 0.01$$

Question 7

Which of the following statements about unbiased estimators is TRUE?

Accepted Answer

The mean of the sampling distribution equals the corresponding population parameter.

Question 8

If the population is not normally distributed, which statement about the sampling distribution of the sample mean is correct when $$n \geq 30$$?

Accepted Answer

The sampling distribution of the sample mean will be approximately normal.

Question 9

Which formula correctly calculates the standard error of the sample mean?

Accepted Answer

$$SE_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$$

Question 10

A researcher wants to know the probability that the sample mean pulse rate is less than 83 bpm, given $$\mu = 78$$, $$\sigma = 13.5$$, and $$n = 10. $$Which $$z-$$score calculation is correct?

Accepted Answer

$$z = \frac{83 - 78}{13.5 / \sqrt{10}}$$

Question 11

Which of the following is NOT a requirement for using the Central Limit Theorem for sample proportions?

Accepted Answer

The sample size $$n$$ must be less than 10.

Question 12

If the population proportion $$p = 0.88$$ and sample size $$n = 1000$$, what is the probability that the sample proportion $$\hat{p} is $$less than $$0.86$$? (Use $$SE_{\hat{p}} = 0.01)$$

Accepted Answer

$$P(z \u003c \frac{0.86 - 0.88}{0.01}) = P(z \u003c -2) \approx 0.0228$$

Question 13

Which of the following best describes the sampling distribution of sample means $$\bar{X}$$?

Accepted Answer

It is the distribution of sample means with all samples of same size $$n$$ taken from the same population.

Question 14

If the sample variance $$s^2 is an $$unbiased estimator of the population variance $$\$$sigma^2$$, what does this mean?

Accepted Answer

The average value of $$s^2$$ over all possible samples equals $$\$$sigma^2$$.

Question 15

Which formula gives the standard error for the sample proportion?

Accepted Answer

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

Question 16

A population is right-skewed. What happens to the sampling distribution of the sample mean as sample size increases?

Accepted Answer

It becomes more normal in shape due to the Central Limit Theorem.

Question 17

Which of the following is a source of bias in estimation?

Accepted Answer

A systematic tendency to over- or under-estimate the true population value.

Question 18

If $$n = 48$$ and the population mean $$\mu = 30$$, what is the expected mean of the sampling distribution of sample means?

Accepted Answer

$$\mu_{\bar{X}} = 30$$

Estimation and the Central Limit Theorem for Proportion & Mean

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