Question

In March 2015, the Public Policy Institute of California (PPIC) surveyed 7525 likely voters living in California. In the survey, respondents were asked about global warming. PPIC researchers are interested in the difference in viewpoints across racial/ethnic groups. PPIC results show that 75% of Latinos view global warming as a serious problem, and 46% of whites view global warming as a serious problem. Using the data from the survey, we calculate the sample difference in global warming viewpoints for Latino respondents and white respondents to be 29% = 0.29. The 95% confidence interval based on this sample difference is (0.275, 0.305). Which of the following is a valid conclusion? Check all that apply. Group of answer choices We are 95% confident that the proportion of Latinos who view global warming as a serious problem is about 27.5% to 30.5% lower than the proportion of whites who view global warming as a serious problem. We are 95% confident that the true difference in proportion of Latinos who view global warming as a serious problem and whites who view global warming as a serious problem is 27.5% to 30.5%. We are 95% confident that the difference between Latino and white opinions about the severity of global warming is 29% with a margin of error of 1.5%.

Answer 1

Answer:

$(\hat p_A -\hat p_B) \pm z_{\alpha/2} \sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A} +\frac{\hat p_B (1-\hat p_B)}{n_B}}$

And for this case the confidence interval is given by: (0.275, 0.305)

We can estimate the proportion difference as:

$\hat p_D = \frac{0.275+0.305}{2}=0.29$

And the margin of error would be:

$ME=0.305-0.29=0.015$

So then for this case the possibl two options are:

We are 95% confident that the true difference in proportion of Latinos who view global warming as a serious problem and whites who view global warming as a serious problem is 27.5% to 30.5%.

We are 95% confident that the difference between Latino and white opinions about the severity of global warming is 29% with a margin of error of 1.5%.

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

$p_A$ represent the real population proportion of Latinos who view global warming as a serious problem

$\hat p_A=0.75$ represent the estimated proportion Latinos who view global warming as a serious problem

$n_A$ is the sample size required of Latinos who view global warming as a serious problem

$p_B$ represent the real population proportion of white who view global warming as a serious problem

$\hat p_B =0.46$ represent the estimated proportion  of whitewho view global warming as a serious problem

$n_B$ is the sample size required of white

$z$ represent the critical value for the margin of error  

Solution to the problem

The population proportion have the following distribution  

$p \sim N(p,\sqrt{\frac{p(1-p)}{n}})$  

The confidence interval for the difference of two proportions would be given by this formula  

$(\hat p_A -\hat p_B) \pm z_{\alpha/2} \sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A} +\frac{\hat p_B (1-\hat p_B)}{n_B}}$

And for this case the confidence interval is given by: (0.275, 0.305)

We can estimate the proportion difference as:

$\hat p_D = \frac{0.275+0.305}{2}=0.29$

And the margin of error would be:

$ME=0.305-0.29=0.015$

So then for this case the possibl two options are:

We are 95% confident that the true difference in proportion of Latinos who view global warming as a serious problem and whites who view global warming as a serious problem is 27.5% to 30.5%.

We are 95% confident that the difference between Latino and white opinions about the severity of global warming is 29% with a margin of error of 1.5%.

Answer 2

Using confidence interval concepts, the correct option is:

We are 95% confident that the difference between Latino and white opinions about the severity of global warming is 29% with a margin of error of 1.5%.

x% confidence interval:

A confidence interval is built from a sample, has bounds a and b, and has a confidence level of x%. It means that we are x% confident that the population mean is between a and b.

In this problem:

The 95% confidence interval based on this sample difference of the proportions of Latinos and Whites is (0.275, 0.305).

• The sample proportion is the mean of these bounds, hence $\overline{p} = \frac{0.275 + 0.305}{2} = 0.29$.
• The margin of error is half the difference, hence: $M = \frac{0.305 - 0.275}{2} = 0.015$

Hence, the correct option is:

We are 95% confident that the difference between Latino and white opinions about the severity of global warming is 29% with a margin of error of 1.5%.

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