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The hypothesis test for proportions helps us determine if a sample proportion is significantly different from a hypothesized population proportion. In this context, we are examining if the proportion of nearsighted children in a sample deviates from the believed 8% in the wider population.

We start by setting up two hypotheses:

• Null Hypothesis (\( H_0 \)): The proportion of nearsighted children is 0.08.
• Alternative Hypothesis (\( H_1 \)): The proportion of nearsighted children is not 0.08. This makes it a two-tailed test, as we are looking for any difference.

By comparing the sample proportion to the hypothesized proportion under these conditions, we can use statistical methods to determine if there's evidence suggesting the assumed proportion isn't accurate.

Calculating the standard error (SE) is crucial in understanding the variability expected in the sample proportion compared to the population proportion. The SE provides a measure of how much sampling variability we might expect. It is derived using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \] Here, \( p \) is the population proportion, \( 0.08 \), and \( n \) is the sample size, 194.

This calculates to about 0.0196, meaning the variability around the sample proportion that we expect due to random sampling is around this value. The smaller the SE, the more accurate our sample proportion is when estimating the true population proportion.

The Z-statistic helps us understand how far the sample proportion is from the hypothesized population proportion, in terms of standard deviations. We calculate it with the formula:

• \( Z = \frac{\hat{p} - p}{SE} \)

Let's break it down:

• \( \hat{p} \) is the sample proportion, which is about 0.1082.
• \( p \) is the population proportion, assumed to be 0.08.
• SE is the calculated standard error, approximately 0.0196.

Substituting these values:

• \( Z = \frac{0.1082 - 0.08}{0.0196} \approx 1.439 \).

This Z-statistic expresses the distance of the sample proportion from the hypothesized proportion in standard error units.

The p-value helps us determine the statistical significance of our results. A p-value is the probability of observing a sample statistic as extreme as the one we obtained, assuming the null hypothesis is true.

Using our calculated Z-statistic of 1.439, we use statistical tables or software to find the p-value in a standard normal distribution. For a two-tailed test:

• If Z equals 1.439, the p-value is around 0.1501.

This p-value indicates the likelihood of obtaining the observed data if the null hypothesis is correct. Since 0.1501 is greater than a typical threshold of 0.05, we do not reject the null hypothesis. Therefore, the data does not provide strong enough evidence to say the real proportion of nearsighted children differs from the supposed 8%.