Exercises - Confidence Intervals for Means (One Sample)

1. For a $90\%$ confidence interval with 10 degrees of freedom, find $t_{\alpha/2}$.

1.812

2. A confidence interval for a mean is $23 \lt \mu \lt 35$. Find the point estimate and the maximum error of the estimate.

$\bar x=29,\ E=6$

3. How large a sample would you need in order to estimate the mean height of women to within 2 cm with $99\%$ confidence? Assume the standard deviation is approximately 9 cm.

For $z_{\alpha/2}=2.576$, the sample size must be at least 135.

4. Suppose you want to estimate the mean number of chocolate chips in a certain brand of chocolate chip cookies. How large a sample would you need in order to estimate the mean within 0.5 chocolate chips with $95\%$ confidence? A pilot study indicates that the standard deviation for the population is around 2.55.

For $z_{\alpha/2}=1.96$, the sample size must be at least 100.

5. Below is a random sample of times for a swimming event. $$\begin{array}{ccccccccc} 154.61&158.03&164.22&165.19&165.64&168.62&170.08&173.17&174.48\\ 175.62&175.82&176.47&176.58&177.68&180.33&183.63&185.71&186.49 \end{array}$$

1. Find a $95\%$ confidence interval for the mean time for this event.

2. How large a sample is needed to find the mean time for the swimming event within 0.2 seconds with $95\%$ confidence?

1. Assumptions: check that there are no outliers and the distribution is not significantly skewed.

$t_{\alpha/2}=2.110,\ E=4.44\qquad 168.5 \lt \mu \lt 177.4$

We are $95\%$ confident the mean time for the swimming event is between 168.5 and 177.4 seconds.

2. For $z_{\alpha/2}=1.96$, $\sigma\approx 8.93$, the sample size must be at least 7659.

6. A sample of apples has weights (in grams) given below. $$\begin{array}{ccccc} 139\qquad 169\qquad 165\qquad 133\qquad 164\cr 150\qquad 163\qquad 137\qquad 132\qquad 134\cr 105\qquad 114\qquad 103\qquad 123\qquad 111 \end{array}$$

1. Find a $95\%$ confidence interval for the mean weight of this kind of apple.

2. How large a sample would be needed to estimate the mean weight of these apples with $99\%$ confidence and margin of error at most 5 grams?

1. Assumptions: check that there are no outliers and the distribution is not significantly skewed.

$t_{\alpha/2}=2.145$
$E=12.4\qquad 123.7 \lt \mu \lt 148.5$

We are $95\%$ confident the mean weight of these apples is between 123.7 and 148.5 grams.

2. For $z_{\alpha/2}=2.576$, $\sigma\approx 22.4$, the sample size must be at least 134.

7. A school nurse wants to determine the average caloric content of the snacks that students are choosing. Below is a random sample of the number of Calories in 10 students' snacks. $$\begin{array}{cccccccccc} 110&125&155&185&190&195&210&235&245&350 \end{array}$$

1. Find a $90\%$ confidence interval for the mean number of calories in the snacks students choose.

2. If the nurse wanted to get a better estimate, how large a sample would be needed in order to estimate to within 10 Calories the mean number of calories with $95\%$ confidence?

1. Assumptions: check that there are no outliers and the distribution is not significantly skewed.

$t_{\alpha/2}=1.833$,
$E=39.6$

$160.4 \lt \mu \lt 239.6$

We are $90\%$ confident the mean number of calories in the snacks is between 160.4 and 239.6.

2. For $z_{\alpha/2}=1.96,\ \sigma\approx 68.3$, the sample size must be at least 180.

8. A study of Oxford College students finds a $90\%$ confidence interval for their mean age at the time of graduation to be from 242.4 to 256.3 months. Which of the following statements can be correctly inferred?

1. $90\%$ of the students in the sample have ages between 242.4 to 256.3 months.

2. $90\%$ of Oxford College students have ages between 242.4 to 256.3 months.

3. There is a $90\%$ probability that a random sample of Oxford College students has a mean age between 242.4 to 256.3 months.

4. There is a $90\%$ probability that the mean age of all Oxford college students is between 242.4 to 256.3 months.

Only (d).