## Related questions with answers

In addition to tests involving biology concepts, students were also tested on process skills. The results of pretest and posttest scores, published in The American Biology Teacher, are given below.

Pretest: All BACC Classes: Mean 10.52, Sample Size 395, Standard Deviation 4.79 Pretest: All Traditional: Mean 11.97, Sample Size 379 and Standard Deviation 5.39 Posttest: All BACC Classes: Mean 14.06, Sample Size 376, Standard Deviation 5.65 Posttest: All Traditional: Mean 12.96, Sample Size 308, Standard Deviation 5.93

a. Find a 95% confidence interval for the mean score on process skills for the posttest for all BACC classes. b. Find a 95% confidence interval for the mean score on process skills for the posttest for all traditional classes. c. Find a 95% confidence interval for the difference in mean scores on process skills for the posttest BACC classes and the posttest traditional classes. d. Does the confidence interval in c provide evidence that there is a real difference in the mean process skills scores between posttest BACC and traditional class scores? Explain.

Solution

Verified(a) Given:

$\begin{align*} n&=\text{Sample size}=376 \\ \overline{x}&=\text{Sample mean}=14.06 \\ s&=\text{Sample standard deviation}=5.65 \\ c&=\text{Confidence coefficient}=95\%=0.95 \end{align*}$

Since the sample size $n$ is large (at least 30), it is appropriate to estimate the population standard deviation by the sample standard deviation.

$\sigma\approx s$

For confidence level $1-\alpha=0.95$, determine $z_{\alpha/2}=z_{0.025}$ using the normal probability table in the appendix (look up 0.025 in the table, the z-score is then the found z-score with opposite sign):

$z_{\alpha/2}=1.96$

The margin of error is then:

$E=z_{\alpha/2}\times \dfrac{\sigma}{\sqrt{n}}\approx z_{\alpha/2}\times \dfrac{s}{\sqrt{n}}=1.96\times \dfrac{5.65}{\sqrt{376}}\approx 0.5711$

The boundaries of the confidence interval then become:

$\overline{x}-E=14.06-0.5711=13.4889$

$\overline{x}+E=14.06+0.5711= 14.6311$

We are 95% confident that the mean score on process skills for the posttest for all BACC classes is between 13.4889 and 14.6311.

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