Problem 2 - Basic Statistics for Business and Economics with Student

Question

A sample of 65 observations is selected from one population with a population standard deviation of \(0.75 .\) The sample mean is \(2.67 .\) A sample of 50 observations is selected from a second population with a population standard deviation of \(0.66 .\) The sample mean is \(2.59 .\) Conduct the following test of hypothesis using the .08 significance level. $$ \begin{array}{l} H_{0}: \mu_{1} \leq \mu_{2} \\ H_{1}: \mu_{1} > \mu_{2} \end{array} $$ a. Is this a one-tailed or a two-tailed test? b. State the decision rule. c. Compute the value of the test statistic. d. What is your decision regarding \(H_{0} ?\) e. What is the \(p\) -value?

Step-by-Step Solution

Answer

a. One-tailed test. b. Reject \(H_0\) if \(z > 1.405\). c. \(z \approx 0.613\). d. Do not reject \(H_0\). e. \(p\)-value \(\approx 0.270\).

Step 1: Determine Test Type

Given the hypotheses: - Null hypothesis, \(H_{0}: \mu_{1} \leq \mu_{2} \) - Alternative hypothesis, \(H_{1}: \mu_{1} > \mu_{2} \) This is a one-tailed test since we are only interested in whether \(\mu_{1}\) is greater than \(\mu_{2}\).

Step 2: Establish Decision Rule

Since this is a one-tailed test with significance level \(\alpha = 0.08\), we determine the critical value from the normal distribution table. For a one-tailed test at \(\alpha = 0.08\), the critical value \(z_{c}\) is approximately 1.405. Decision Rule: Reject \(H_0\) if the test statistic \(z > 1.405\).

Step 3: Compute the Test Statistic

The test statistic for the difference between two independent means is given by:\[ z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]Where:- \(\bar{x}_1 = 2.67\), \(\bar{x}_2 = 2.59\)- \(\sigma_1 = 0.75\), \(n_1 = 65\)- \(\sigma_2 = 0.66\), \(n_2 = 50\)Substitute the values:\[ z = \frac{2.67 - 2.59}{\sqrt{\frac{0.75^2}{65} + \frac{0.66^2}{50}}} \]\[ z \approx \frac{0.08}{\sqrt{0.00862 + 0.008712}} \]\[ z \approx \frac{0.08}{0.130476} \]\[ z \approx 0.613 \]

Step 4: Decision on \(H_0\)

Compare the computed test statistic (\(z = 0.613\)) to the critical value (\(z_c = 1.405\)). Since 0.613 is less than 1.405, we do not reject \(H_0\).

Step 5: Determine the p-value

For \(z = 0.613\) in a standard normal distribution, the p-value is the probability of observing a test statistic as extreme as 0.613 in the right tail.Using normal distribution tables or software, the p-value is about 0.270.Since the p-value (0.270) is greater than \(\alpha = 0.08\), it further confirms that we do not reject \(H_0\).

Key Concepts

One-Tailed Test

A one-tailed test focuses on identifying if there is a significant increase or decrease in a dataset in a specific direction. In the context of hypothesis testing, we use a one-tailed test when we are only interested in a single direction of effect. Here, our alternative hypothesis suggests that the mean of the first population \(\mu_{1}\) is greater than the mean of the second population \(\mu_{2}\). Thus, we are only looking at the possibility of an increase, which directly points to a one-tailed test dealing with the right tail of a distribution.

• Right-tailed test: Tests if a value is significantly greater than the hypothesized value, as seen here with \(H_1: \mu_1 > \mu_2\).
• Left-tailed test: Opposite of the right-tailed test, it checks for significantly smaller values.

Knowing which type of test to use—one-tailed or two-tailed—helps us correctly interpret results and avoid incorrect conclusions.

Significance Level

The significance level, denoted by \(\alpha\), plays a crucial role in hypothesis testing. It represents the probability threshold for rejecting the null hypothesis. In our example, we used a significance level of 0.08. Essentially, this means we were willing to accept an 8% chance of incorrectly rejecting the null hypothesis.

• Higher \(\alpha\) values: Increases the risk of Type I error (false positive).
• Lower \(\alpha\) values: Decreases the same risk but might increase Type II error (false negative).

Choosing the right \(\alpha\) is essential as it balances these two types of errors, depending on the context and necessity of the study. In this way, decisions are made more confidently, knowing the margin of error is considered and controlled.

Test Statistic

A test statistic is a calculated value that we obtain from our sample data during hypothesis testing. It helps us to determine how far off our sample results are from the null hypothesis. This computation involves several elements derived from the sample, such as the mean and standard deviation, and is mainly dependent on the type of test.
For our exercise, to find the test statistic, the formula for the comparison of two means is used:\[ z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]Each component of this formula is carefully derived from the exercise, enabling us to reach a test statistic, which quantifies the difference between the two dataset means.

• Value interpretation: The computed statistic (\(z = 0.613\) in this case) tells us how many standard deviations our sample mean difference is from the population mean specified in the null hypothesis.

The test statistic fundamentally determines if our observed effect is sizeable enough to question the null hypothesis.

P-Value

The p-value is a crucial component in making decisions about hypotheses. It quantifies the strength of the evidence against the null hypothesis. Specifically, it indicates the probability of observing a test statistic as extreme as the one observed if the null hypothesis were true. In our example, the computed p-value is 0.270.

• Low p-value (< \(\alpha\)): Strong evidence against the null hypothesis, leading to its rejection.
• High p-value (> \(\alpha\)): Weak evidence against the null hypothesis, suggesting no rejection.

Here, our p-value is higher than the significance level (0.270 > 0.08). This confirms our earlier decision not to reject the null hypothesis. The result implies our observed data, and any variations could very likely occur under the null hypothesis. Thus, p-value determination is vital for concluding if the observed data indeed differ from expected norms or could occur by random chance.