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The concept of normal distribution is fundamental in statistics. It describes how the values of a variable are likely to be distributed in a population. Picture a bell-shaped curve—this is the graphical representation of a normal distribution. It is symmetric, with most data points clustered around the central peak and fewer points at the extremes.

In our exercise, the mean income of plumbers was stated to follow this type of distribution. Key features of a normal distribution are its mean and standard deviation:

• Mean (\( \mu \)): The average of all values. Here, it's \( \\(45,000 \).
• Standard Deviation (\( \sigma \)): This measures the spread of the values around the mean. Smaller values mean data points are closer to the mean, while larger values indicate a wider spread. In our case, it's \( \\)3,000 \).

Understanding these helps in determining the shape and spread of the income data among plumbers.

In hypothesis testing, determining whether to use a one-tailed or two-tailed test is crucial. A two-tailed test checks for the possibility of an effect in both directions, either above or below the mean.

For our scenario, the alternative hypothesis is that the mean income of plumbers is "not equal to" \( \$45,000 \). This is why a two-tailed test is appropriate. We're not just interested in whether the mean is greater or lesser, but simply whether it's different.

• If we thought the mean could only increase, a one-tailed test might be used instead.
• The significance level determines how often we're willing to be wrong if the null hypothesis is actually true. Here, it is set at 0.10.

Using a two-tailed test accounts for uncertainty in both directions, confirming robust statistical testing through this approach.

A key outcome of hypothesis testing is the p-value, which tells us how likely we are to observe the data we have, assuming the null hypothesis is true.

In simpler terms, it's the probability of obtaining results at least as extreme as the ones observed during the test. You calculate and interpret this value to decide whether or not to reject the null hypothesis.

• A smaller p-value means stronger evidence against the null hypothesis. Typical thresholds are 0.05 or 0.10.
• In our example, the p-value is approximately 0.068, which is less than the significance level of 0.10, indicating enough evidence to reject the null hypothesis.

This process ultimately helps confirm that changes or differences in data are statistically significant, not just due to random chance.

The Z-test is used to determine if there's a significant difference between sample and population means. It is ideal when the population variance is known and the sample size is large.

The formula for the Z-test statistic plays a major role:\[Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]Here, each component has a specific purpose:

• \(\bar{x}\): Sample mean, which in this case is \(\\(45,500\).
• \(\mu\): Population mean, set at \(\\)45,000\).
• \(\sigma\): Standard deviation, which is \(\$3,000\).
• \(n\): Sample size, given as 120.

After substituting the values, we calculate the Z value to be 1.826.

This calculated Z-value is then compared to the critical Z-values to determine whether the sample mean significantly deviates from the population mean. For our two-tailed test at 0.10 significance, the critical Z-value is \( \pm 1.645 \), showing that our calculated Z is indeed significant.