1. [C10] The number of terminals in use by Group Alfa during...

1. [C10] The number of terminals in use by Group Alfa during the lunch hour at Fairlady Aircraft is normally distributed. A random sample of 40 observed values of the number of active terminals is shown in the table. Assume $\sigma=5$. At the $5 \%$ level of significance, test the null hypothesis that the mean number of active terminals $\mu=20$ against the alternative hypothesis that $\mu>20$. $$ \begin{array}{rrrr} {\text { Random Sample }} \\ \hline & & & \\ 22.5585 & 16.7495 & 19.8800 & 19.8130 \\ 22.3250 & 23.2865 & 15.7555 & 11.8815 \\ 24.5805 & 22.1430 & 30.7650 & 24.0120 \\ 23.1480 & 16.2845 & 21.1555 & 22.1095 \\ 19.0270 & 18.8885 & 24.2645 & 21.9145 \\ 26.7180 & 27.4775 & 22.8960 & 14.3475 \\ 18.3270 & 10.4450 & 27.1350 & 11.1425 \\ 23.0950 & 26.8640 & 18.0160 & 9.9325 \\ 21.5260 & 27.2705 & 21.5315 & 20.2230 \\ 9.0565 & 18.7445 & 24.9890 & 17.7345 \\ \hline \end{array} $$

1. [C10] The number of terminals in use by Group Alfa during the lunch hour at Fairlady Aircraft is normally distributed. A random sample of 40 observed values of the number of active terminals is shown in the table. Assume $\sigma=5$. At the $5 \%$ level of significance, test the null hypothesis that the mean number of active terminals $\mu=20$ against the alternative hypothesis that $\mu>20$.
$$
\begin{array}{rrrr}
{\text { Random Sample }} \\
\hline & & & \\
22.5585 & 16.7495 & 19.8800 & 19.8130 \\
22.3250 & 23.2865 & 15.7555 & 11.8815 \\
24.5805 & 22.1430 & 30.7650 & 24.0120 \\
23.1480 & 16.2845 & 21.1555 & 22.1095 \\
19.0270 & 18.8885 & 24.2645 & 21.9145 \\
26.7180 & 27.4775 & 22.8960 & 14.3475 \\
18.3270 & 10.4450 & 27.1350 & 11.1425 \\
23.0950 & 26.8640 & 18.0160 & 9.9325 \\
21.5260 & 27.2705 & 21.5315 & 20.2230 \\
9.0565 & 18.7445 & 24.9890 & 17.7345 \\
\hline
\end{array}
$$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide between two competing hypotheses about a population parameter based on sample data. It involves setting up a null hypothesis, which represents a default position or a statement of no effect, and an alternative hypothesis, which represents the claim being tested.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around its mean, with data tapering off equally to both sides. It is characterized by its bell-shaped curve, and when data are normally distributed, many statistical methods, including hypothesis tests, can be applied reliably due to well-known properties of the distribution.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement about a population parameter that indicates no significant effect or difference, while the alternative hypothesis (H?) suggests that a significant effect or difference exists. In testing hypotheses for a mean, one typically sets the null hypothesis with a specified value and the alternative in relation to that value (e.g., greater than, less than, or not equal to).

Significance Level

The significance level, commonly denoted by ?, is the probability threshold for rejecting the null hypothesis. It defines the risk of committing a Type I error, which occurs when the null hypothesis is incorrectly rejected. A common significance level used in hypothesis tests is 5%, meaning there is a 5% risk of a false positive.

Z-Test

The Z-test is a type of hypothesis test used when the population standard deviation is known and the data are assumed to be normally distributed. It involves comparing the sample mean to the hypothesized population mean using a test statistic that follows a standard normal distribution under the null hypothesis.

Standard Error

The standard error measures the variability of the sample mean as an estimate of the population mean, calculated by dividing the population standard deviation by the square root of the sample size. It quantifies the precision of the sample mean and is a crucial component when calculating the test statistic in hypothesis testing.

Test Statistic and Decision Rule

The test statistic is a numerical value calculated from the sample data that is compared to a critical value or used to obtain a p-value in order to decide whether to reject the null hypothesis. In the context of a Z-test, the test statistic is computed by subtracting the hypothesized mean from the sample mean and dividing by the standard error. The decision rule is based on whether this statistic falls into the rejection region defined by the significance level.

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