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 Y and Gen 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 Y and Gen 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

α = 0.01.

(Use a statistical computer package to calculate the P-value. Use p_Y − p_X. Round your test statistic to two decimal places and your P-value to three decimal places.)

z	=
P-value	=

State your conclusion.

We fail to reject H₀. We have convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion of those in Gen X.We reject H₀. We don't have convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion of those in Gen X.    We fail to reject H₀. We don't have convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion of those in Gen X.We reject H₀. We have convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion of those in Gen X.

(b) Estimate the difference between the proportion of Gen Y and the proportion of Gen X that made a donation via text message using a 99% confidence interval. (Round your answers to three decimal places.)
(  ,  )

Provide an interpretation of the interval.

We are 99% confident that the true difference between the proportion of Gen Y and the proportion of Gen X who made a donation via text message is between these two values.

We are 99% confident that the true difference between the proportion of Gen Y and the proportion of Gen X who made a donation via text message is directly in the middle of these two values.  

  We are 99% confident that the true proportion of Gen X is between these two values.

We are 99% confident that the true proportion of Gen Y is between these two values.

We are 99% confident that the true proportion of Gen Y is directly in the middle of these two values.

Provide an interpretation of the associated confidence level.

In repeated sampling with a sample size of 400, 99% of the resulting confidence intervals would contain the true proportion of Gen X who donated via text message.

In repeated sampling with any sample size, 99% of the resulting confidence intervals would contain the true difference in proportions who donated via text message.  

In repeated sampling with a sample size of 400, 99% of the resulting confidence intervals would contain the true proportion of Gen Y who donated via text message.

In repeated sampling with a sample size of 400, 99% of the resulting confidence intervals would contain the true difference in proportions who donated via text message

In repeated sampling with any sample size, 99% of the resulting confidence intervals would contain the true proportion of Gen X who donated via text message.

Answer 1

Answer a)

z = 1.17

P-value corresponding to z = 1.172 for a right tailed test is P(Z > 1.172). P-value is obtained using the calculator.

P-value = 0.1205

Since, p-value = 0.1205 > α = 0.01, we fail to reject null hypothesis.

Conclusion: We fail to reject H0. We don't have convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion of those in Gen X.

Answer b)

Step 1: Find α/2
Level of Confidence = 99%
α = 100% - (Level of Confidence) = 1%
α/2 = 0.5% = 0.005


Step 2: Find zα/2
Calculate zα/2 by using standard normal distribution (normal population with mean (μ) = 0 and standard deviation (σ) = 1)
with α/2 = 0.005 as right-tailed area and left-tailed area.

zα/2 = 2.5758

Step 3: Calculate Confidence Interval

Lower Bound = (p̂1 - p̂2) - zα/2•√p̂1(1 - p̂1)/n1+ p̂2(1 - p̂2)/n2 = -0.036
Upper Bound = (p̂1 - p̂2) + zα/2•√p̂1(1 - p̂1)/n1+ p̂2(1 - p̂2)/n2 = 0.096

99% Confidence Interval = (-0.036, 0.096)

Interpretation

We are 99% confident that the true difference between the proportion of Gen Y and the proportion of Gen X who made a donation via text message is between these two values.

In repeated sampling with a sample size of 400, 99% of the resulting confidence intervals would contain the true difference in proportions who donated via text message.