Problem 28 - Basic Statistics for Business and Economics with Student

Question

Each month the National Association of Purchasing Managers publishes the NAPM index. One of the questions asked on the survey to purchasing agents is: Do you think the economy is expanding? Last month, of the 300 responses, 160 answered yes to the question. This month, 170 of the 290 responses indicated they felt the economy was expanding. At the .05 significance level, can we conclude that a larger proportion of the agents believe the economy is expanding this month?

Step-by-Step Solution

Answer

We cannot conclude a larger proportion believes the economy is expanding this month.

Step 1: Identify Hypotheses

We start with the null and alternative hypotheses. The null hypothesis \( H_0 \) states that the proportion of agents who think the economy is expanding this month is the same or less than last month, i.e., \( p_2 \leq p_1 \). The alternative hypothesis \( H_a \) asserts that a larger proportion of agents think the economy is expanding this month, i.e., \( p_2 > p_1 \).

Step 2: Define Sample Proportions

Calculate the sample proportions for each month. For last month, \( \hat{p}_1 = \frac{160}{300} = 0.5333 \). For this month, \( \hat{p}_2 = \frac{170}{290} = 0.5862 \).

Step 3: Compute Pooled Proportion

The pooled proportion \( \hat{p} \) is calculated as a weighted average of the two proportions:\[ \hat{p} = \frac{160 + 170}{300 + 290} = \frac{330}{590} = 0.5593 \]

Step 4: Calculate Standard Error

Determine the standard error using the pooled proportion:\[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.5593 \cdot (1 - 0.5593) \cdot \left( \frac{1}{300} + \frac{1}{290} \right)} \approx 0.0438 \]

Step 5: Find Test Statistic

Calculate the test statistic \( z \), which compares the difference in sample proportions to the standard error:\[ z = \frac{\hat{p}_2 - \hat{p}_1}{SE} = \frac{0.5862 - 0.5333}{0.0438} \approx 1.206 \]

Step 6: Determine Critical Value and Conclusion

At a significance level of \( \alpha = 0.05 \) for a one-tailed test, the critical z-value is 1.645. Since the calculated z-value (1.206) is less than 1.645, we fail to reject the null hypothesis.

Key Concepts

Significance Level

In hypothesis testing, the significance level plays a crucial role. It is denoted by \( \alpha \), and it represents the probability of rejecting the null hypothesis when it is actually true. Essentially, it's the threshold for deciding whether the observed result is significant.
Choosing a significance level is not just about picking any number; it reflects how much risk of a Type I error (false positive) one is willing to accept. Commonly used levels are 0.05, 0.01, and 0.10. The lower the \( \alpha \), the stricter the criteria for rejecting the null hypothesis.

• For instance, with an \( \alpha = 0.05 \), there’s a 5% chance of incorrectly concluding there is an effect when there isn’t one.
• This level is chosen depending on the context and consequences of making a Type I error.

In the exercise above, the significance level is 0.05, meaning the test is considered significant if the probability of the observed difference under the null hypothesis is less than 5%.
This is why the calculated z-value is compared with the critical value derived from the \( \alpha \), to determine whether or not to reject the null hypothesis.

Sample Proportions

Sample proportions serve as estimates of the actual proportion of the population, based on the sample data collected. They are expressed as \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes or desired outcomes in the sample, and \( n \) is the sample size.
In the provided exercise, the sample proportions for last month and this month were calculated to understand opinions on economic expansion.

• The first month's sample proportion was \( \hat{p}_1 = \frac{160}{300} = 0.5333 \).
• The second month's sample proportion was \( \hat{p}_2 = \frac{170}{290} = 0.5862 \).

These proportions indicate the respective percentages of agents who answered "yes" to believing in economic expansion.
This calculation helps to discern any change over time, by comparing proportions from different samples.
Sample proportions are vital for carrying out hypothesis tests like the one in this exercise, allowing us to establish whether an observed effect is statistically significant.

Standard Error

The standard error (SE) is a measure of the variability or dispersion of a sample statistic, like the sample mean or sample proportion. It tells us how much the sample statistic would vary, if we were to take multiple samples from the same population.
In hypothesis testing, SE is particularly important as it serves as the denominator in the test statistic formula, thus influencing how likely we are to see differences due to random sampling variability.

• It is derived from the pooled proportion \( \hat{p} = \frac{330}{590} = 0.5593 \) in this problem.
• The formula used is: \[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]
• Upon calculation, \( SE \approx 0.0438 \).

This value quantifies the amount of variation we can expect in the difference between the two sample proportions, purely due to chance.
The standard error is then used to calculate the z-test statistic, which assesses the observed difference relative to this expected variation.
Therefore, understanding the standard error is essential for interpreting hypothesis test results accurately.