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When conducting hypothesis testing, one common approach is the two-tailed test. This type of test is used when you want to determine if there is a significant difference in either direction from a proposed value. In the exercise, the null hypothesis is that the population mean \( \mu = 95 \). We are testing against an alternative hypothesis that \( \mu eq 95 \). This means we suspect that the true mean could be either greater than or less than 95.

For a two-tailed test, the critical region is located in both tails of the normal distribution. Here, the test is performed at a significance level of \( \alpha = 0.01 \), meaning that the total area of rejection is distributed evenly between both tails, with each tail containing 0.005 of the area. This setup is crucial because it affects where the critical z-values lie; it is these values that the calculated z-score is compared against to determine significance. If the z-score falls outside of the critical values, the null hypothesis can be rejected.

In hypothesis testing, understanding errors is vital. Type I errors occur when the null hypothesis is rejected when it is true. In the context of the exercise, a Type I error would mean concluding that the mean melting point of the oil is different from 95 when it is not. The probability of this happening is denoted by \( \alpha \), which is 0.01 here. This translates into being 1% willing to take the risk of incorrectly rejecting the null hypothesis.

On the other hand, a Type II error happens when the null hypothesis is not rejected when it should have been. In our example, it would mean failing to detect that the mean melting point is indeed different from 95 when it really is. This probability is denoted by \( \beta \). In relationship to the problem, the second part requires calculating \( \beta \) when \( \mu = 94 \). The result helps determine the test's sensitivity and is found to be approximately 0.816, indicating an 81.6% probability of not detecting a true difference of 1 degree if that is the actual case.

The z-score is a statistical measure that describes a value's position relative to the mean of a group of values. It is measured in terms of standard deviations from the mean. For hypothesis testing, the z-score helps determine how far away our sample result is from what we would expect under the null hypothesis.

In the given problem, the z-score is calculated using the formula: \[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \]Here, \( \bar{x} = 94.32 \), \( \mu = 95 \), \( \sigma = 1.20 \), and \( n = 16 \). Substituting these values gives a z-score of \(-2.27\). This tells us that the sample mean is 2.27 standard deviations below the hypothesized population mean of 95. However, this z-score is not in the critical region (i.e., does not exceed \( \pm 2.576 \)), suggesting that the sample provides insufficient evidence to reject the null hypothesis at the 0.01 level.

Determining the appropriate sample size for an experiment or survey is crucial for obtaining reliable results. In hypothesis testing, the sample size needs to be large enough to detect a true effect when there is one, while controlling for Type II errors.

For the exercise, you are tasked with finding the sample size \( n \) that would ensure \( \beta = 0.1 \) when the true mean \( \mu_a \) is 94 with \( \alpha = 0.01 \). This involves using the formula:\[ n = \left(\frac{(z_{\alpha/2} + z_{\beta}) \sigma}{\mu_0 - \mu_a}\right)^2 \]Where \( z_{\beta} \approx 1.28 \) for \( \beta = 0.1 \). Calculating this gives a necessary sample size of approximately 81. This means that to achieve a low risk of not detecting a real difference (Type II error of 10%), 81 samples must be analyzed. Increasing sample size generally enhances the power of a test, reducing the chance of Type II errors.