Q: If a sample of 225 students has a mean age of 20.9 years and the population standard deviation is 2 years, what is the 95% confidence interval for the population mean?

$$20.9 \pm 1.96 \times \frac{2}{\sqrt{225}}$$

Question 1

Suppose the average weight of women is 140 lbs with a standard deviation of 10 lbs. If a random sample of 25 women is selected, what is the probability that their average weight is between 139.4 lbs and 141.2 lbs? (Assume weights are normally distributed.)

Accepted Answer

$$0.3360$$

Question 2

Which of the following is a correct application of the Central Limit Theorem (CLT)?

Accepted Answer

The distribution of sample means will be approximately normal if the sample size is large, regardless of the population distribution.

Question 3

Given a population mean $$\mu = 20$$ and a sample mean $$\bar{x} = 20.9$$ with a standard error of $$1.25$$, what is the $$z-$$score for the sample mean?

Accepted Answer

$$z = \frac{20.9 - 20}{1.25} = 0.72$$

Question 4

If the standard deviation of a population is $$\sigma = 10$$ and the sample size is $$n = 25$$, what is the standard error of the mean?

Accepted Answer

$$\frac{10}{\sqrt{25}} = 2$$

Question 5

A 95% confidence interval for a population mean is calculated as $$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}. $$What is the value of $$z_{\alpha/2}$$ for a 95% confidence level?

Accepted Answer

$$1.96$$

Question 6

If a sample of 225 students has a mean age of 20.9 years and the population standard deviation is 2 years, what is the 95% confidence interval for the population mean?

Accepted Answer

$$20.9 \pm 1.96 	imes \frac{2}{\sqrt{225}}$$

Question 7

Which of the following statements about hypothesis testing is correct?

Accepted Answer

The null hypothesis is rejected if the test statistic falls in the critical region.

Question 8

If the sample mean is $$\bar{x} = 141.2$$, population mean $$\mu = 140$$, population standard deviation $$\sigma = 10$$, and sample size $$n = 25$$, what is the $$z-$$score?

Accepted Answer

$$z = \frac{141.2 - 140}{10/\sqrt{25}} = 0.60$$

Question 9

Which of the following best describes the meaning of a 95% confidence interval for a population mean?

Accepted Answer

There is a 95% probability that the interval contains the true population mean.

Question 10

A researcher wants to test if the average age of students is greater than 20. Which is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu \leq 20$$

Question 11

If the calculated $$z-$$score for a sample mean is $$0.72$$, which of the following is the correct interpretation?

Accepted Answer

The sample mean is $$0.72$$ standard errors above the population mean.

Question 12

Which formula is used to calculate the confidence interval for a population mean when the population standard deviation is known?

Accepted Answer

$$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

Central Limit Theorem, Confidence Intervals, and Hypothesis Testing: Study Notes

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