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A two-tailed test is used when we are interested in determining whether a sample mean is significantly different from a hypothesized population mean. Unlike one-tailed tests that look for an increase or decrease, the two-tailed approach checks both possibilities.
In hypothesis testing, we start by stating two hypotheses: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_1 \)). For this exercise, the null hypothesis is \( H_0: \mu = 155 \), which suggests the mean melting point is 155°F. The alternative hypothesis is \( H_1: \mu eq 155 \), indicating any significant deviation from this value.

• Critical values are determined by the significance level (\( \alpha \)), where results that fall outside this range lead to rejection of \( H_0 \).
• A two-tailed test checks both ends of the distribution, allowing for increased detection of any difference from the hypothesized mean.

The Z-test is a statistical method used to determine if there is a significant difference between sample and population means. It is valid when the population standard deviation is known and the sample size is large (typically \( n > 30 \)), or when the distribution is normally distributed even with smaller samples.
To calculate the Z-test statistic, we use the formula: \[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \] where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

• The Z-statistic quantifies the distance between the sample mean and the population mean in terms of standard deviation units.
• In our scenario, the Z-value was calculated to be approximately \( -1.688 \).
• If the calculated Z-value falls beyond the critical values determined by \( \alpha \), it implies statistical significance and leads to rejection of \( H_0 \).

The P-value is an essential concept in hypothesis testing. It helps us understand the strength of our results. Simply put, it tells us the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis \( H_0 \) is true.
The P-value is used to determine whether to reject \( H_0 \). In our exercise, the P-value for the calculated Z-statistic of \( -1.688 \) is derived from standard normal distribution tables or calculators.

• A lower P-value (< \( \alpha \)) suggests that the null hypothesis should be rejected.
• If the P-value is \( 0.0914 \) as calculated, it indicates that there is a 9.14% chance of observing such data if \( H_0 \) is true. This is above the \( \alpha = 0.01 \) threshold, so \( H_0 \) is not rejected.
• The decision process is easier by comparing the P-value directly with \( \alpha \).

Determining the right sample size is crucial to ensure reliable test results. A sample that is too small may not accurately represent the population, while a sample that is unnecessarily large could be a waste of resources. For hypothesis testing, particularly when wanting to control both \( \alpha \) and \( \beta \) errors, sample size calculation is key.
In this exercise, the goal is to find the minimum sample size (\( n \)) needed to ensure a \( \beta \)-error less than 0.1 when the true mean \( \mu \) is 150 with a significance level \( \alpha \) of 0.01. Formula used: \[ n = \left( \frac{Z_{\text{critical}} + Z_{\text{power}}}{\Delta / \sigma} \right)^2 \] where \( \Delta \) is the target difference.

• \( Z_{\text{critical}} = 2.576 \) for \( \alpha = 0.01 \).
• \( Z_{\text{power}} \approx 1.282 \) for \( \beta = 0.1 \).
• Calculated minimum sample size was approximately \( 6 \), reflecting more precision is needed for smaller targets.