Q: Which of the following is the general form of a confidence interval for a mean?

point estimate $$\pm ($$values from tables) $$\times ($$standard error)

Question 1

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

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

Question 2

A sample of 50 houses has a mean water meter fitting cost of €115 and a standard deviation of €14.5. What is the 95% confidence interval for the population mean cost?

Accepted Answer

$$110.98,\ 119.02$$

Question 3

If the population standard deviation $$\sigma is $$known, which formula should be used for a 95% confidence interval for the population mean?

Accepted Answer

$$\bar{X} \pm 1.96 \frac{\sigma}{\sqrt{n}}$$

Question 4

Which of the following will decrease the sampling error when estimating a population mean?

Accepted Answer

Increasing the sample size

Question 5

A sample of 25 cafes in Limerick has a mean cappuccino price of €3.85 and a standard deviation of €0.53. Using a $$t-$$value of 2.064, what is the 95% confidence interval for the mean price?

Accepted Answer

$$3.63,\ 4.07$$

Question 6

When should the Student $$t-$$distribution be used instead of the normal distribution for confidence intervals?

Accepted Answer

When the sample size is less than 30 and the population standard deviation is unknown

Question 7

A weight-loss company claims an average loss of 4kg per month. A sample of 50 customers has a mean loss of 3.5kg and a standard deviation of 1.4kg. The 95% confidence interval is (3.11, 3.89). What does this interval suggest?

Accepted Answer

The company's claim is not supported by the data.

Question 8

What is the standard error of the mean for a sample of size $$n$$ with standard deviation $$s$$?

Accepted Answer

$$\frac{s}{\sqrt{n}}$$

Question 9

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A 95% confidence interval means that if we repeated sampling, 95% of such intervals would contain the true population mean.

Question 10

If you want a 99% confidence interval instead of a 95% confidence interval, what should you do?

Accepted Answer

Use 2.58 instead of 1.96 as the critical value.

Question 11

Which of the following is the general form of a confidence interval for a mean?

Accepted Answer

point estimate $$\pm ($$values from tables) $$	imes ($$standard error)

Question 12

A bag of 100 gold coins is weighed to estimate the mean coin weight. If the population mean is 50g and standard deviation is 5g, what is the standard error of the mean?

Accepted Answer

$$\frac{5}{\sqrt{100}} = 0.5$$

Question 13

Which of the following is NOT a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

There is a 95% probability that the true population mean lies within the interval calculated from this sample.

Question 14

Why does the Student $$t-$$distribution have heavier tails than the standard normal distribution?

Accepted Answer

To account for the extra uncertainty when estimating the population standard deviation from a small sample.

Question 15

If two groups have 95% confidence intervals for their means that do not overlap, what does this suggest?

Accepted Answer

There is a statistically significant difference between the group means.

Question 16

For a sample of size $$n=10$$, what is the appropriate $$t-$$value for a 95% confidence interval (using $$v = n-1$$ degrees of freedom)?

Accepted Answer

2.262

Question 17

Which of the following factors will increase the width of a confidence interval for a population mean?

Accepted Answer

Increasing the sample standard deviation

Question 18

What is the main reason for using a sample mean $$\bar{x} to $$estimate a population mean $$\mu$$?

Accepted Answer

It provides a point estimate of the unknown population mean.

L9 Confidence Intervals for a Population Mean: Central Limit Theorem, Z and t-Distributions

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