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Question

# 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 5.93a. 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.

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(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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