Problem 1 - Basic Statistics for Business and Economics with Student

Question

(a) Is this a one- or two-tailed test? (b) What is the decision rule? (c) What is the value of the test statistic? (d) What is your decision regarding \(H_{0} ?\) (e) What is the \(p\) -value? Interpret it. The following information is available. \(H_{0}: \mu=50\) \(H_{i}: \mu \neq 50\) The sample mean is \(49,\) and the sample size is \(36 .\) The population standard deviation is 5\. Use the .05 significance level.

Step-by-Step Solution

Answer

This is a two-tailed test. Do not reject \( H_0 \); the p-value is 0.2302.

Step 1: Determine the Type of Test

Since the null hypothesis is \( H_0: \mu = 50 \) and the alternative hypothesis is \( H_i: \mu eq 50 \), this suggests that any deviation from 50 is significant. Therefore, this is a two-tailed test because we are looking for significance in both directions (less than 50 and greater than 50).

Step 2: Establish the Decision Rule

For a two-tailed test with a significance level of \( \alpha = 0.05 \), we divide this significance level by two to account for both tails, resulting in \( 0.025 \) per tail. Using a standard normal distribution (Z-distribution), we find the critical values corresponding to \( 0.025 \) on each side. The Z-critical values for a two-tailed test at \( \alpha = 0.05 \) are approximately \( -1.96 \) and \( 1.96 \). Thus, if the calculated test statistic falls outside the range of \( -1.96 \) to \( 1.96 \), we reject the null hypothesis.

Step 3: Compute the Test Statistic

To find the test statistic, use the Z-test formula: \[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]where \( \bar{x} = 49 \) (sample mean), \( \mu = 50 \) (hypothesized population mean), \( \sigma = 5 \) (population standard deviation), and \( n = 36 \) (sample size).Calculate:\[ Z = \frac{49 - 50}{\frac{5}{\sqrt{36}}} = \frac{-1}{\frac{5}{6}} = \frac{-1}{0.8333} \approx -1.20 \]

Step 4: Make the Decision Regarding \( H_0 \)

The calculated test statistic is \( -1.20 \). Since \( -1.20 \) falls within the range of \(-1.96\) to \(1.96\), we do not reject the null hypothesis. This means there is not enough evidence to suggest that the population mean is different from 50.

Step 5: Calculate and Interpret the p-value

To determine the p-value, we evaluate the probability of observing a test statistic as extreme or more extreme than \( -1.20 \) in a standard normal distribution. Using a Z-table or calculator, we find that the p-value for \( Z = -1.20 \) is approximately 0.1151 for one tail; therefore, for two tails, it is \( 0.1151 \times 2 = 0.2302 \). Since the p-value \( 0.2302 \) is greater than the significance level of \( 0.05 \), it supports our decision not to reject the null hypothesis. The result is not statistically significant.

Key Concepts

Two-tailed Test

In hypothesis testing, a two-tailed test is employed when we are interested in detecting a significant difference in either direction from the hypothesized parameter. This means we do not just look for values that are greater or less than a specific point, but for deviations in both directions.
The null hypothesis in our example states that the population mean is 50, denoted as \( H_0: \mu = 50 \). The alternative hypothesis, \( H_i: \mu eq 50 \), suggests any mean different from 50 is significant. In a two-tailed test, this covers the possibility of the mean being either less than or more than 50.

• We split the significance level \( \alpha \) between two tails of the distribution.
• The two-tailed test checks for extremes on both ends of the normal distribution.
• It is a more comprehensive approach since it considers both higher and lower deviations as potential evidence against the null hypothesis.

This kind of test is particularly useful when there's no specific directional expectation.

Decision Rule

A decision rule in hypothesis testing dictates when to reject the null hypothesis. For a two-tailed test, you first determine the critical value or range of values.

• **Significance Level**: The significance level, \( \alpha = 0.05 \), is divided equally for two-tailed tests, meaning each tail has an area of \( 0.025 \).
• **Z-distribution**: We use the Z-distribution to find critical values at \( \alpha = 0.025 \) for each tail, which means looking up the Z-table to get approximately \(-1.96\) and \(1.96\) as critical values.
• **Decision Making**: If the calculated test statistic is less than \(-1.96\) or greater than \(1.96\), the null hypothesis is rejected.'

This rule is crucial because it provides a clear boundary to make statistical decisions. Falling within the range implies there is insufficient evidence to reject \( H_0 \).

Test Statistic

The test statistic is a standardized value that provides a basis for decision making within hypothesis testing. It comes from the data and allows comparison against the critical value(s).In our example, we compute the Z-test statistic using the formula:\[Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]Here's how it works:

• \( \bar{x} = 49 \), the sample mean.
• \( \mu = 50 \), the population mean (under null hypothesis).
• \( \sigma = 5 \), the population standard deviation.
• \( n = 36 \), the sample size.

By plugging in the numbers, you find:\[Z = \frac{49 - 50}{\frac{5}{6}} = \frac{-1}{0.8333} \approx -1.20\]Since \(-1.20\) does not exceed the critical boundaries \(-1.96, 1.96\), this statistic fails to provide enough evidence to reject \( H_0 \). It shows how the sample mean stands in relation to the hypothesized mean.

Significance Level

The significance level, denoted as \( \alpha \), plays a pivotal role in hypothesis testing by defining the threshold for deciding whether an observed effect is statistically significant.In this exercise, \( \alpha = 0.05 \) means there is a 5% risk of rejecting the null hypothesis if it is actually true. Here's why it's important:

• **Threshold Setting**: Determines the region(s) of the distribution where the null hypothesis will be rejected. For a two-tailed test, this is split into \( 0.025 \) for each tail.
• **Risk Management**: Represents the probability of a Type I error, where a true \( H_0 \) is incorrectly rejected. Lower \( \alpha \) values reduce this risk but could increase Type II error chances, missing a true effect.
• **p-value Comparison**: It provides a benchmark against which the p-value is compared. If the p-value is less than \( \alpha \), the result is statistically significant.

Understanding the significance level's function helps in assessing how data-driven decisions can potentially affect conclusions from hypothesis testing.