91Ó°ÊÓ

Shipping holiday gifts. A December 2010 survey asked 500 randomly sampled Los Angeles residents which shipping carrier they prefer to use for shipping holiday gifts. The table below shows the distribution of responses by age group as well as the expected counts for each cell (shown in parentheses). Method \begin{tabular}{l|cc|cc|cc|c} & \multicolumn{6}{|c|} { Age } & \\ \cline { 2 - 5 } & \multicolumn{2}{|c|} {\(18-34\)} & \multicolumn{2}{|c|} {\(35-54\)} & \multicolumn{2}{|c|} {\(55+\)} & Total \\ \hline USPS & 72 & \((\mathrm{~B} 1)\) & 97 & (102) & 76 & (62) & 245 \\ UPS & 52 & (83) & 76 & \((6 \mathrm{~b})\) & 34 & (41) & 162 \\ FedEx & 31 & (21) & 24 & (27) & 9 & (16) & 64 \\ Something else & 7 & (5) & 6 & (7) & 3 & (4) & 16 \\ Not sure & 3 & (5) & 6 & (5) & 4 & (3) & 13 \\ \hline Total & \multicolumn{2}{|c|} {165} & \multicolumn{2}{|c|} {209} & \multicolumn{2}{|c|} {126} & 500 \end{tabular} (a) State the null and alternative hypotheses for testing for independence of age and preferred shipping method for holiday gifts among Los Angeles residents. (b) Are the conditions for inference using a chi-square test satisfied?

Step by step solution

01

Define the Hypotheses

The null hypothesis \(H_0\) is that there is no association between age group and preferred shipping method. The alternative hypothesis \(H_a\) is that there is an association between age group and preferred shipping method. Formally, \(H_0: \text{Age and shipping preference are independent}\) and \(H_a: \text{Age and shipping preference are not independent}\).

02

Check for Expected Counts

To use the chi-square test, all expected counts should be at least 5. Checking the table: USPS (all expected counts \(\geq 5\)), UPS (all expected counts \(\geq 5\)), FedEx (all expected counts \(\geq 5\)), Something else (all expected counts \(\geq 5\)), Not sure (all expected counts \(\geq 5\)). All expected counts are indeed 5 or more.

03

Verifying Sample Size

The sample size should be large enough, generally 5 times the number of expected categories. There are 5 age categories and a total of 500 responses, which meets this requirement, ensuring sufficient data for the chi-square test.

04

Conclusion: Conditions for Chi-Square Test

Since all expected frequencies are greater than or equal to 5, and the sample size is ample, the conditions for performing a chi-square test are satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis is a fundamental concept in statistics and hypothesis testing. It's a statement that suggests no effect or no relationship between two measured phenomena. In the context of our problem, the null hypothesis (\(H_0\)) proposes that age group and preferred shipping method are independent of each other.

In simpler terms, this means if the null hypothesis is true, knowing a person's age would give you no information about their shipping preference. We often start with this hypothesis because it represents a baseline or default position that there is no change or effect to observe.

Testing the null hypothesis involves trying to find evidence to the contrary. If enough data suggests otherwise, we can reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

The alternative hypothesis is the complementary statement to the null hypothesis. In statistical analyses, it proposes that there is an effect or a relationship present. For the exercise at hand, the alternative hypothesis (\(H_a\)) suggests that there is indeed a relationship between age group and preferred shipping method.

This means that age does influence shipping preference.

• If the null hypothesis is rejected, it leads us to support the alternative hypothesis.
• Evidence against the null could indicate that different age groups favor different shipping methods for sending holiday gifts.

The alternative hypothesis is what researchers often hope to prove – indicating a meaningful connection or pattern they are investigating.

Expected Counts

Expected counts are a crucial component in conducting a chi-square test. They are used to determine what the counts would be if the null hypothesis were indeed true, meaning no association between the variables. In this case, expected counts help us figure out the distribution of shipping preferences across different age groups, assuming there is no preference based on age.

To calculate the expected count for each cell, use the formula:\[\text{Expected Count} = \frac{\text{(Row Total)} \times \text{(Column Total)}}{\text{(Overall Total)}}\]This creates a distribution of expected counts across the table, which is then used to compare against the observed counts using the chi-square test.

• We need each expected count to be at least 5 to ensure the validity of the test results.
• Comparing the observed counts with these expected values helps us determine whether to reject the null hypothesis.

Independence Test

An independence test, such as the chi-square test, is used to explore whether two categorical variables are related. More specifically, it assesses whether the distribution of one variable differs depending on the level of another variable.

In this scenario, the independence test seeks to determine if there is a statistical relationship between age groups and shipping method preferences.

• The chi-square test compares the observed counts of responses with the expected counts.
• It calculates a chi-square statistic, which measures how much the observed counts diverge from the expected counts.
• If the calculated statistic is significantly high, this suggests a dependency or relationship between the variables.

This test is especially useful in social sciences and market research, providing insights into patterns and association between different groups or preferences.