Question

After the 2010 earthquake in Haiti, many charitable organizations conducted fundraising campaigns to raise money for emergency relief. Some of these campaigns allowed people to donate by sending a text message using a cell phone to have the donated amount added to their cell-phone bill. The report "Early Signals on Mobile Philanthropy: Is Haiti the Tipping Point?" (Edge Research, 2010 ) describes the results of a national survey of 1526 people that investigated the ways in which people made donations to the Haiti relief effort. The report states that \(17 \%\) of Gen \(Y\) respondents (those born between 1980 and 1988 ) and \(14 \%\) of Gen \(X\) respondents (those born between 1968 and 1979 ) said that they had made a donation to the Haiti relief effort via text message. The percentage making a donation via text message was much lower for older respondents. The report did not say how many respondents were in the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) samples, but for purposes of this exercise, suppose that both sample sizes were 400 and that it is reasonable to regard the samples as representative of the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) populations. a. Is there convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion for Gen X? Use \(\alpha=.01\). b. Estimate the difference between the proportion of Gen \(\mathrm{Y}\) and the proportion of Gen \(\mathrm{X}\) that made a donation via text message using a \(99 \%\) confidence interval. Provide an interpretation of both the interval and the associated confidence level.

Short answer

Without exact calculations, the steps imply a z-score calculation for a hypothesis test and a confidence interval for the difference. The final conclusions will depend on these calculations. The p-value will tell if there's convincing evidence about a difference in text donations between Gen Y and Gen X and the confidence interval will estimate this difference.

Step-by-step solution

Calculate the sample proportions

First, calculate the sample proportions from the given percentages. The sample sizes for both generations are 400. For Gen Y, \(17\%\) donated via text, so \(p_1 = 0.17\). Similarly, for Gen X, \(14\%\) donated via text, so \(p_2 = 0.14\). These are the proportions of successes (donations via text) for each generation.

Set up the hypothesis

The null hypothesis is that the proportions are equal, \(p_1 = p_2\), and the alternative hypothesis is that the proportion of Gen Y is greater than Gen X, \(p_1 > p_2\). The significance level is \(\alpha = 0.01\).

Calculate the pooled proportion

For a two-proportion z test, a combined or pooled proportion of successes is calculated. This is done by dividing the total number of successes by the total sample size. So, \(p = \frac{p_1 \times n_1 + p_2 \times n_2}{n_1 + n_2}\). Here, \(n_1\) and \(n_2\) both are 400, and \(p_1\) and \(p_2\) are 0.17 and 0.14 respectively. So, \(p = \frac{0.17 \times 400 + 0.14 \times 400}{400 + 400}\).

Calculate the z-score

The z-score is calculated using the formula, \(z = \frac{p_1 - p_2}{\sqrt{p(1-p)(\frac{1}{n_1}+\frac{1}{n_2})}}\). The numerator is the difference in sample proportions and the denominator is the standard error of the difference. After substituting \(p_1=0.17\), \(p_2=0.14\), \(n_1=400\), \(n_2=400\), and \(p\) (calculated earlier), calculate for \(z\).

Conclude the hypothesis test

Refer the z-score to a standard normal (z) distribution table or use a calculator to find the p-value. If the p-value ≤ \(\alpha\), then reject the null hypothesis. If the p-value > \(\alpha\), then we do not reject the null hypothesis. Our result will tell us whether there is significant evidence to claim that a greater proportion of Gen Y donated via text compared to Gen X.

Estimate the difference between the proportions

The \(99\%\) confidence interval for the difference between the proportions of donations from Gen Y and Gen X (\(p_1 - p_2\)) is given by \((p_1 - p_2) ± z_{0.005} * \sqrt{(\frac{{p_1(1-p_1)}}{{n_1}}) + (\frac{{p_2(1-p_2)}}{{n_2}})}\). Here, \(z_{0.005}\) is the critical value for a \(99\%\) confidence interval. Calculate the difference taking \(p_1 = 0.17\), \(p_2 = 0.14\), \(n_1 = 400\) and \(n_2 = 400\).

Interpret the result

The interval estimates the difference in proportions for the entire population of Gen Y and Gen X. If this interval includes 0, then we can say there's no significant difference between the two. If it doesn't include 0, then we can say there's a significant difference. The \(99\%\) confidence level means that we can be \(99\%\) confident that the true population difference falls within this interval.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistical inference used to determine if there is enough evidence to support a certain claim about a population. In this context, the exercise questions whether the proportion of Gen Y who donated to Haiti relief via text message is greater than that for Gen X. Here’s how this process works:

The first step involves setting up two hypotheses:

• The Null Hypothesis (H₀): There is no difference in the donation proportions, thus \(p_1 = p_2\).
• The Alternative Hypothesis (Hₐ): Gen Y has a greater proportion of donors, thus \(p_1 > p_2\).

We then define the significance level, \(\alpha = 0.01 \), which indicates the probability of rejecting a true null hypothesis. A lower \(\alpha\) means stricter criteria for evidence.

The hypothesis test uses a two-proportion z-test, given the categorical data from two independent groups. The calculated z-score and corresponding p-value tell us if our sample provides sufficient evidence to reject \(H₀\).

If the p-value is less than \(\alpha\), we reject \(H₀\). In this exercise, a conclusive result shows if Gen Y notably donated more than Gen X via text.

Confidence Intervals

Confidence intervals provide a range of values that estimate a parameter with a certain level of confidence. In this case, the exercise asks for a 99% confidence interval for the difference in donation proportions between Gen Y and Gen X. Here, the aim is to gather not just a single possibility but a spectrum of plausible differences.

To calculate the confidence interval, the difference in sample proportions \(p_1 - p_2\) is first determined. The standard error of this difference helps determine the width of the interval:

• A critical value, \(z_{0.005}\), is used which corresponds to the desired confidence level (99% means a 1% risk of error split across both tails of the normal distribution).
• The formula is: \[(p_1 - p_2) \pm z_{0.005} \times \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\]

This calculation gives us upper and lower bounds, representing the range within which the true difference in population proportions likely lies. If 0 is not within this interval, it signifies a meaningful difference between the groups, as assumed for this exercise.

Proportions Testing

Proportions testing is an essential method when dealing with categorical data from different groups, assessing if there's a significant difference in specific proportions. In the exercise, it focuses on testing whether the proportion of Gen Y donating via text is greater than Gen X.

Key steps involve:

• Determining each group's proportion of successes—here, text donations (\(p_1 = 0.17\) for Gen Y and \(p_2 = 0.14\) for Gen X).
• A pooled proportion: Since we are comparing two groups, a combined proportion \(p\) is calculated, accounting for potential variance across samples.
• Using this pooled proportion to compute the standard error, facilitating the z-score calculation.

The two-proportion z-test is then employed, aiming to evaluate if the evidence is substantial enough to claim that one group’s proportion exceeds the other’s. The focus here on statistical evidence helps in understanding population-level differences through sample data.