Problem 21 - Basic Statistics for Business and Economics with Student

Question

The owner of a mail-order catalog would like to compare her sales with the geographic distribution of the population. According to the United States Bureau of the Census, 21 percent of the population lives in the Northeast, 24 percent in the Midwest, 35 percent in the South, and 20 percent in the West. Listed below is a breakdown of a sample of 400 orders randomly selected from those shipped last month. At the .01 significance level, does the distribution of the orders reflect the population?

Step-by-Step Solution

Answer

The distribution of orders does not differ significantly from the population distribution.

Step 1: State the Hypotheses

The null hypothesis (H_0) states that the order distribution matches the population distribution, i.e., there are no differences significant enough to cause concern. The alternative hypothesis (H_1) proposes that there is a difference. More formally: \[H_0: \text{The order distribution matches the population distribution.}\]\[H_1: \text{The order distribution does not match the population distribution.}\]

Step 2: Collect and Summarize the Data

The given data for the percentage breakout of sales orders is the following:- Northeast: 92 orders- Midwest: 108 orders- South: 123 orders- West: 77 ordersConvert these counts to proportions based on a sample of 400 orders:Northeast: \( \frac{92}{400} = 0.23 \) Midwest: \( \frac{108}{400} = 0.27 \) South: \( \frac{123}{400} = 0.3075 \) West: \( \frac{77}{400} = 0.1925 \)

Step 3: Expected Frequencies Under Null Hypothesis

Using the population percentages, calculate the expected numbers of orders for each region under the null hypothesis:- Northeast: \( 0.21 \times 400 = 84 \)- Midwest: \( 0.24 \times 400 = 96 \)- South: \( 0.35 \times 400 = 140 \)- West: \( 0.20 \times 400 = 80 \)

Step 4: Calculate the Chi-Square Statistic

Use the chi-square formula:\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]Where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency:- Northeast: \( \frac{(92 - 84)^2}{84} = 0.7619 \)- Midwest: \( \frac{(108 - 96)^2}{96} = 1.5 \)- South: \( \frac{(123 - 140)^2}{140} = 2.1857 \)- West: \( \frac{(77 - 80)^2}{80} = 0.1125 \)Total \( \chi^2 = 0.7619 + 1.5 + 2.1857 + 0.1125 = 4.5601 \)

Step 5: Determine the Critical Value and Make a Decision

At a significance level of 0.01 and with 3 degrees of freedom (4 regions - 1), the critical value from the chi-square distribution table is approximately 11.345. Compare the calculated chi-square value (4.5601) with the critical value. Since 4.5601 < 11.345, we fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves making an initial assumption, called the null hypothesis, and then testing this assumption using statistical methods. If the data provides sufficient evidence against the null hypothesis, it can be rejected in favor of the alternative hypothesis. In this context, the null hypothesis is that the geographic distribution of orders matches the distribution of the population. This means that the observed orders are consistent with what we'd expect based on population percentages. The alternative hypothesis suggests a difference between the order distribution and the population distribution. Hypothesis testing is important to determine whether observed differences are statistically significant or just due to random chance, and is often used in various fields such as scientific research, economics, and social sciences.

Significance Level

The significance level in statistical hypothesis testing is a threshold used to determine whether a result is statistically significant. Often denoted by alpha (\(\alpha\)), it represents the probability of rejecting the null hypothesis when it is actually true—essentially the risk of a false positive. A common significance level used in research is 0.05, but in this exercise, a more stringent significance level of 0.01 is applied. This means that there is only a 1% chance of incorrectly rejecting the null hypothesis. By setting a significance level, researchers control how much evidence they require before deciding there is a real effect or difference. In hypothesis testing, if the p-value, calculated from the data, is less than the significance level, the null hypothesis is rejected, indicating the result is statistically significant.

Expected Frequencies

Expected frequencies are calculated based on the assumption that the null hypothesis is true. For each category or group in the dataset, expected frequencies give the number of observations that should occur if the null hypothesis holds true. They're calculated using the formula \(E_i = \text{Total number of orders} \times \text{Proportion of population in each category}\). For example, if 21% of the population lives in the Northeast, the expected number of orders from that region is calculated by multiplying 21% by the total number of orders. In this exercise, expected frequencies help us determine the degree of difference between what we observe and what we'd expect if the population distribution correctly predicted order distribution. Comparing expected and observed frequencies is crucial in calculating the chi-square statistic, which tests whether the differences are significant.

Observed Frequencies

Observed frequencies are the actual counts or numbers of occurrences measured in the sample data. In the exercise, this refers to the number of orders from each geographic region—Northeast, Midwest, South, and West. These are real figures collected from a sample of 400 orders and are compared with expected frequencies to assess if there is a significant departure from what was expected. Through calculating the differences between observed and expected frequencies and using those in the chi-square formula, we obtain a chi-square statistic that indicates how well the observed data fits the expected distribution under the null hypothesis. Observed frequencies are foundational to hypothesis testing as they provide the empirical data needed to draw statistical conclusions.