Q: Suppose you want a 95% confidence interval for the mean $$x of a $$normal population with known standard deviation $$\sigma. $$What is the formula for the margin of error?

$$\frac{1.96 \times \sigma}{\sqrt{n}}$$

Q: Which formula would you use to calculate the required sample size $$n$$ for a desired margin of error $$E in $$estimating a population mean with known standard deviation $$\sigma at $$confidence level $$z^*$$?

$$n = \left( \frac{z^* \times \sigma}{E} \$$right)^2$$

Question 1

Which of the following best describes the margin of error in a confidence interval for the mean of a normal population?

Accepted Answer

It only covers random sampling errors and ignores other sources of error.

Question 2

A researcher wants to estimate the mean blood pressure in a population using a simple random sample (SRS). Which condition must be met for the confidence interval to be valid?

Accepted Answer

The variable must have an exactly Normal distribution in the population.

Question 3

Suppose you want a 95% confidence interval for the mean $$x of a $$normal population with known standard deviation $$\sigma. $$What is the formula for the margin of error?

Accepted Answer

$$\frac{1.96 	imes \sigma}{\sqrt{n}}$$

Question 4

Which of the following statements about statistical significance is correct?

Accepted Answer

Statistical significance depends on both the effect size and the sample size.

Question 5

In hypothesis testing, what does a Type I error represent?

Accepted Answer

Rejecting the null hypothesis when it is true.

Question 6

A medical study tests whether a new drug lowers cholesterol. The null hypothesis is that the drug has no effect. If the study finds a statistically significant result, what does this mean?

Accepted Answer

There is strong evidence against the null hypothesis, but practical importance must be considered.

Question 7

Which of the following increases the power of a statistical test?

Accepted Answer

Increasing the significance level $$\alpha.$$

Question 8

If you want to detect a small effect in a population with high confidence, what should you do?

Accepted Answer

Increase the sample size.

Question 9

Which formula would you use to calculate the required sample size $$n$$ for a desired margin of error $$E in $$estimating a population mean with known standard deviation $$\sigma at $$confidence level $$z^*$$?

Accepted Answer

$$n = \left( \frac{z^* 	imes \sigma}{E} \$$right)^2$$

Question 10

A matched pairs experiment is conducted to test for loss of sweetness in a food product. The test statistic follows a normal distribution with mean difference $$\mu$$ and standard deviation $$\sigma. $$What does a higher effect size imply?

Accepted Answer

A greater chance of rejecting the null hypothesis if the alternative is true.

Question 11

Which of the following is a practical problem that can affect the validity of a confidence interval?

Accepted Answer

Under-coverage and nonresponse.

Question 12

If the variable you measure does not have an exactly Normal distribution, which of the following is true about inference for the mean?

Accepted Answer

Inference may still be valid if the sample size is large due to the Central Limit Theorem.

Question 13

What is the purpose of a test of significance in statistics?

Accepted Answer

To describe how convincing the evidence provided by the sample is against the null hypothesis.

Question 14

Which of the following best describes a Type II error?

Accepted Answer

Failing to reject the null hypothesis when it is false.

Question 15

A company wants to be 90% confident of detecting a mean loss of 0.8 points in sweetness on a 10-point scale. What statistical concept is most relevant to this scenario?

Accepted Answer

Power of a statistical test.

Question 16

Which of the following is NOT a factor that affects the required sample size for a study?

Accepted Answer

The color of the data points.

Question 17

If you want to reduce the probability of a Type I error in your hypothesis test, what should you do?

Accepted Answer

Decrease the significance level $$\alpha.$$

Question 18

Which of the following is true about the P-value in a one-sided test compared to a two-sided test?

Accepted Answer

The P-value for a one-sided test is half the P-value for the two-sided test.

Question 19

Why is it important to consider practical significance in addition to statistical significance?

Accepted Answer

A statistically significant result may not be meaningful in real-world terms.

Statistical Inference: Confidence Intervals, Significance Tests, and Power

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