Question

In one income group, \(45 \%\) of a random sample of people express approval of a product. In another income group, \(55 \%\) of a random sample of people express approval. The standard errors for these percentages are \(.04\) and \(.03\) respectively. Test at the \(10 \%\) level of significance the hypothesis that the percentage of people in the second income group expressing approval of the product exceeds that for the first income group.

Short answer

The null hypothesis (\(H_0\)) states that the percentage of people in the second income group expressing approval of the product does not exceed that for the first group, while the alternative hypothesis (\(H_a\)) states the opposite. To test these hypotheses, we calculate the test statistic \(Z = \frac{0.55 - 0.45}{\sqrt{0.04^2 + 0.03^2}}\). Then, with a 10% level of significance, we find the critical value 1.28 for a one-tailed test. Comparing the test statistic with the critical value allows us to determine whether to reject or not reject the null hypothesis.

Step-by-step solution

Setting up the Hypotheses

The null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) are set up like this: \(H_0: \mu_2- \mu_1 \leq 0\) (The percentage of people in the second income group expressing approval of the product does not exceed that for the first group) \(H_a: \mu_2- \mu_1 > 0\) (The percentage of people in the second income group expressing approval of the product exceeds that for the first group) In our case, \(\mu_1\) and \(\mu_2\) would be the population proportions of those approve the product in the first and second income group, respectively.

Calculation of The Test Statistic

The test statistic for this hypothesis test is a z score, obtains as follows: \[Z = \frac{(\hat{p}_2 - \hat{p}_1) - (p_2 - p_1)}{\sqrt{{se}_1^2 + {se}_2^2}}\] Here, \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions (0.45 and 0.55) and \({se}_1\) and \({se}_2\) are the standard errors of the two samples (0.04 and 0.03). Assuming under \(H_0\), \(p_2 - p_1 \leq 0\), we have \[Z = \frac{0.55 - 0.45}{\sqrt{0.04^2 + 0.03^2}}\]

Calculation of the Critical Value

The critical value is the point beyond which we would reject the null hypothesis. Because the alternative hypothesis posits that the proportion of approvers in the second group is greater than in the first group, this is a one-tailed test. At a 10% level of significance, the critical value of z, in this case, is 1.28. This critical value is typically found using a z-table or statistical software.

Decision Making

Compare the test statistic with the critical value to make a decision. If the calculated test statistic (Z score) is greater than the critical value, reject the null hypothesis in favor of the alternative hypothesis. If the calculated Z is less than the critical value, do not reject the null hypothesis. Assuming you've done the calculation for Z in step two, you can now make the decision.

Key concepts

Null Hypothesis

The null hypothesis, symbolized as \(H_0\), is the default or baseline assumption in hypothesis testing. It's a statement which we assume to be true unless we find convincing evidence to the contrary. In the context of comparing two populations, the null hypothesis often asserts that there is no difference in the characteristics of interest between the groups. For instance, when examining approval of a product across different income groups, the null hypothesis would usually claim there's no difference in approval rates.

A critical aspect is that \(H_0\) is what we are testing against. It's not what we are trying to prove but rather what we are trying to refute with our test. The goal is to see if the data provide strong enough evidence to reject the null hypothesis in favor of the alternative.

Alternative Hypothesis

On the flip side, the alternative hypothesis, represented as \(H_a\) or \(H_1\), is the statement that reflects a change, effect, or difference. When conducting our test on product approval, the alternative hypothesis posits that there is a difference in approval rates between the two income groups. Specifically in our example, it suggests that the second income group has a higher approval rate than the first one.

It's the hypothesis that researchers often want to find evidence for, but we can only lend it credibility if we can show the null hypothesis is unlikely to be true. The strength of this evidence is determined through the rest of the hypothesis testing process.

Test Statistic

The test statistic is a standardized value that is calculated from our sample data during the hypothesis testing process. It's essentially a score that measures how far our sample statistic is from the null hypothesis's claimed value considering the assumed standard deviation of the sampling distribution.

In our product approval example, we are using the \(Z\) score as the test statistic to measure the difference in approval rates between the two groups. This \(Z\) score will be compared to a critical value to decide whether the null hypothesis can be rejected.

Significance Level

The significance level, often denoted by \(\alpha\), is a threshold that determines when we should reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it's actually true. A common \(\alpha\) value is 0.05 or 5%, which indicates that there's a 5% risk of rejecting the null hypothesis by mistake.

In our current scenario, the significance level has been set at 10%, which is relatively high, meaning we are willing to accept a larger risk of a Type I error. This higher level can make it easier to reject \(H_0\), but at the same time, it makes our conclusion less rigorous.

Z Score

The \(Z\) score is a statistical measure that tells us how many standard deviations an element is from the mean. In the context of hypothesis testing, the \(Z\) score of the test statistic reveals how extreme the observed difference is assuming the null hypothesis is true. It is calculated by taking the difference between the sample statistic and the null hypothesis value and dividing it by the standard error.

When we calculate the \(Z\) score, we can compare it to critical \(Z\) values that correspond to our chosen significance level. If the \(Z\) score lies beyond that critical value, the result is statistically significant, and we may reject the null hypothesis.

Population Proportion

The population proportion, in our specific example, refers to the true percentage of individuals in a certain population who approve of a product. We don't know this true percentage, so we use sample proportions (such as the 45% for the first income group and 55% for the second) as estimates. These sample proportions are denoted by \(\hat{p}\) and are used to help us infer about the population proportion.

Whether we're looking at income groups, consumers of a product, or any subgroup, the concept applies the same: we're trying to make an educated guess about a proportion within a larger population based on our sample data.

Standard Error

The standard error is a statistic that measures the variability or spread of a sample distribution. In hypothesis testing, particularly when we're estimating population proportions, the standard error helps us understand how much we'd expect our sample proportions to vary purely by chance.

In the given exercise, the standard errors for the approval percentages are 0.04 for the first group and 0.03 for the second. These values are used to quantify the uncertainty associated with our estimates of the population proportions, and they are crucial in calculating our test statistic, the \(Z\) score.