Chapter 8.7, Problem 79E

Refer to Exercise 8.66.

a Another similar study is to be undertaken to compare the mean posttest scores for BACC and traditionally taught high school biology students. The objective is to produce a 99% confidence interval for the true difference in the mean posttest scores. If we need to sample an equal number of BACC and traditionally taught students and want the width of the confidence interval to be 1.0, how many observations should be included in each group?

b Repeat the calculations from part (a) if we are interested in comparing mean pretest scores.

c Suppose that the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for BACC and traditionally taught biology students. If her objective is that both intervals have widths no larger than 1 unit, what sample sizes should be used?

8.66 Historically, biology has been taught through lectures, and assessment of learning was accomplished by testing vocabulary and memorized facts. A teacher-devoloped new curriculum, Biology: A Community Content (BACC), is standards based, activity oriented, and inquiry centered. Students taught using the historical and new methods were tested in the traditional sense on biology concepts that featured biological knowledge and process skills. The results of a test on biology concepts were published in The American Biology Teacher and are given in the following table.¹¹

a Give a 90% confidence interval for the mean posttest score for all BACC students.

b Find a 95% confidence interval for the difference in the mean posttest scores for BACC and traditionally taught students.

c Does the confidence interval in part (b) provide evidence that there is a difference in the mean posttest scores for BACC and traditionally taught students? Explain.

To determine

Find the number of observations that need to be included in each group when the objective is to produce a 99% confidence interval for the true difference in the mean posttest scores.

Answer to Problem 79E

The number of observations that need to be included in each group is 3,007.

Explanation of Solution

Here, ${\sigma }_1=8.03,{\sigma }_2=6.96,\alpha =0.01,\text{length}=1$.

From Table 4: Normal Curve Areas, the z value corresponding to 0.0049 (which is approximately 0.005 $\left(=\frac{1-0.99}{2}\right)$) is 2.58. That is, $z_{\frac{0.01}{2}}=2.58$.

The number of observations that need to be included in each group is computed as follows:

$\begin{array}{c}\text{length}=2\left(z_{\frac{\alpha }{2}}\sqrt{\frac{{\sigma }_1^2}{n}+\frac{{\sigma }_2^2}{n}}\right)\\ 1=2\left[\left(2.58\right)\sqrt{\frac{{8.03}^2}{n}+\frac{{6.96}^2}{n}}\right]\\ 1=2\left[\left(2.58\right)\sqrt{\frac{112.9225}{n}}\right]\\ \sqrt{n}=2\left[\left(2.58\right)\left(10.6265\right)\right]\\ n={54.8327}^2\\ =3,006.625\\ \approx 3,007\end{array}$

Therefore, the number of observations that need to be included in each group is 3,007.

To determine

Find the number of observations that need to be included in each group when the objective is to produce a 99% confidence interval for the true difference in the mean pretest scores.

Answer to Problem 79E

The number of observations that need to be included in each group is 1,623.

Explanation of Solution

Here, ${\sigma }_1=5.59,{\sigma }_2=5.45,\alpha =0.01,\text{length}=1$.

The number of observations that need to be included in each group is computed as follows:

$\begin{array}{c}1=2\left[\left(2.58\right)\sqrt{\frac{{5.59}^2}{n}+\frac{{5.45}^2}{n}}\right]\\ \sqrt{n}=2\left[\left(2.58\right)\left(7.8071\right)\right]\\ n={40.2846}^2\\ =1,622.849\\ \approx 1,623\end{array}$

Therefore, the number of observations that need to be included in each group is 1,623.

To determine

Find the sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for BACC and traditionally taught biology students.

Answer to Problem 79E

The sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for BACC is 2,549.

The sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for traditionally taught biology students is 2,081.

Explanation of Solution

For BACC:

For this case, ${\sigma }_1=5.59,{\sigma }_2=8.03,\alpha =0.01,\text{length}=1$.

The sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for BACC is computed as follows:

$\begin{array}{c}1=2\left[\left(2.58\right)\sqrt{\frac{{5.59}^2}{n}+\frac{{8.03}^2}{n}}\right]\\ \sqrt{n}=2\left[\left(2.58\right)\left(9.7841\right)\right]\\ n={50.4860}^2\\ =2,548.84\\ \approx 2,549\end{array}$

Therefore, the sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for BACC is 2,549.

For traditionally taught:

For this case, ${\sigma }_1=5.45,{\sigma }_2=6.96,\alpha =0.01,\text{length}=1$.

The sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for traditionally taught biology students is computed as follows:

$\begin{array}{c}1=2\left[\left(2.58\right)\sqrt{\frac{{5.45}^2}{n}+\frac{{6.96}^2}{n}}\right]\\ \sqrt{n}=2\left[\left(2.58\right)\left(8.8399\right)\right]\\ n={45.6139}^2\\ =2,080.63\\ \approx 2,081\end{array}$

Therefore, the sample size that is required if the researcher wants to construct 99% confidence intervals to compare both pretest and posttest scores for traditionally taught biology students is 2,081.