Q: Suppose you are constructing a confidence interval for the mean IQ score of statistics students. You want 95% confidence that your sample mean is within 3 points of the population mean, and you estimate $$\sigma = 15. $$What is the minimum sample size required?

$$n = \left[\frac{1.96 \times 15}{3}\right]^2 = 97$$

Question 1

Which of the following is the best point estimate for the population mean $$\mu$$ when using sample data?

Accepted Answer

The sample mean $$\bar{x}$$

Question 2

A researcher wants to construct a 95% confidence interval for the mean cholesterol level in adults, but the population standard deviation $$\sigma is $$unknown. Which formula should be used for the margin of error $$E$$?

Accepted Answer

$$E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

Question 3

Which of the following conditions must be satisfied to use the Student $$t$$ distribution for constructing a confidence interval for the population mean $$\mu$$ when $$\sigma is $$unknown?

Accepted Answer

The sample is a simple random sample and the population is normally distributed or $$n \u003e 30$$

Question 4

A sample of size $$n = 15 is $$taken from a normally distributed population. What is the number of degrees of freedom used to find the critical value $$t_{\alpha/2}$$ for a confidence interval?

Accepted Answer

$$n - 1 = 14$$

Question 5

Suppose you are constructing a confidence interval for the mean IQ score of statistics students. You want 95% confidence that your sample mean is within 3 points of the population mean, and you estimate $$\sigma = 15. $$What is the minimum sample size required?

Accepted Answer

$$n = \left[\frac{1.96 	imes 15}{3}ight]^2 = 97$$

Question 6

Which statement correctly describes the Student $$t$$ distribution as the sample size $$n$$ increases?

Accepted Answer

It approaches the standard normal distribution

Question 7

Given a confidence interval for $$\mu of (120$$, 140), what is the point estimate and margin of error?

Accepted Answer

Point estimate: $$\mu = 130$$, Margin of error: $$E = 10$$

Question 8

A medical researcher wants to estimate the mean recovery time after surgery. The population standard deviation $$\sigma is $$known to be 8 days. What formula should be used for the margin of error $$E$$?

Accepted Answer

$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

Question 9

Which of the following is NOT a valid method for estimating the population standard deviation $$\sigma$$ when determining sample size?

Accepted Answer

Using the population mean $$\mu as an $$estimate for $$\sigma$$

Question 10

A sample of $$n = 22$$ pulse rates is obtained. To construct a 95% confidence interval for the population standard deviation, what is the number of degrees of freedom and which distribution is used?

Accepted Answer

Degrees of freedom: $$21$$, Distribution: Chi-square

Question 11

Which of the following is a correct confidence interval format for estimating a population mean $$\mu$$?

Accepted Answer

$$\bar{x} - E \u003c \mu \u003c \bar{x} + E$$

Question 12

If a population is normally distributed and $$\sigma is $$unknown, which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

Student $$t$$ distribution

Question 13

A researcher wants to be 99% confident that her estimate of the standard deviation $$\sigma of $$pulse rates is within 5% of the true value. What is the minimum sample size required?

Accepted Answer

1336

Question 14

Which statement about the chi-square distribution is TRUE?

Accepted Answer

It is skewed to the right and only takes non-negative values

Question 15

When constructing a confidence interval for the population variance $$\$$sigma^2$$, which formula should be used?

Accepted Answer

$$\frac{(n-1)s^2$}{\$$chi^2$$_R} \u003c \$$sigma^2$$ \u003c \frac{(n-1)s^2$}{\$$chi^2$$_L}$$

Question 16

A sample of size $$n = 10 is $$taken from a population. The sample mean is $$\bar{x} = 50$$ and the sample standard deviation is $$s = 8. $$What is the margin of error $$E$$ for a 95% confidence interval if $$t_{0.025, 9} = 2.262$$?

Accepted Answer

$$E = 2.262 \cdot \frac{8}{\sqrt{10}} = 5.72$$

Question 17

Which of the following is a reason to use a conservative estimate for $$\sigma$$ when determining sample size?

Accepted Answer

It ensures the sample size is not too small, reducing the risk of inadequate estimation

Question 18

A confidence interval for the mean is calculated as (45, 55). What is the interpretation of this interval at the 95% confidence level?

Accepted Answer

We are 95% confident that the true population mean $$\mu$$ lies between 45 and 55

Estimating Parameters and Determining Sample Sizes: Confidence Intervals for Population Mean

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