Answered

Consider the following data:

Survey A asked 1000 people how they liked the new movie "Avengers: Endgame," and 88% said they enjoyed it.

Survey B also concluded that 82% of people liked the movie, but they asked 1600 total moviegoers.

Which of the following is true about this comparison?

A. The margin of error is the same for both.
B. The confidence interval is smaller for Survey B.
C. Increasing the number of people asked does not change the 95% confidence interval.
D. Survey A is more accurate since the percentage is higher.
E. Survey A has more approvals than Survey B.
F. Survey A is better than Survey B since it has a higher percentage.



Answer :

GinnyAnswer
Let's analyze the given surveys step by step.

Step 1: Calculate the standard error for each survey.

The formula for the standard error of proportion (SE) is:
[tex]\[ SE = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
where [tex]\( p \)[/tex] is the proportion of positive responses and [tex]\( n \)[/tex] is the sample size.

For Survey A:
- Sample size, [tex]\( n_A = 1000 \)[/tex]
- Proportion, [tex]\( p_A = 0.88 \)[/tex]

So, the standard error for Survey A ([tex]\( SE_A \)[/tex]) is:
[tex]\[ SE_A = \sqrt{\frac{0.88 \times (1 - 0.88)}{1000}} = 0.010276186062932104 \][/tex]

For Survey B:
- Sample size, [tex]\( n_B = 1600 \)[/tex]
- Proportion, [tex]\( p_B = 0.82 \)[/tex]

So, the standard error for Survey B ([tex]\( SE_B \)[/tex]) is:
[tex]\[ SE_B = \sqrt{\frac{0.82 \times (1 - 0.82)}{1600}} = 0.009604686356149274 \][/tex]

Step 2: Calculate the 95% confidence interval (CI) for each survey.
The formula for the confidence interval is:
[tex]\[ CI = \left( p - z \times SE, p + z \times SE \right) \][/tex]
where [tex]\( z \)[/tex] is the Z-value for the desired confidence level (1.96 for 95%).

For Survey A:
[tex]\[ CI_A = \left( 0.88 - 1.96 \times 0.010276186062932104, 0.88 + 1.96 \times 0.010276186062932104 \right) \][/tex]
[tex]\[ CI_A = (0.859858675316653, 0.900141324683347) \][/tex]

For Survey B:
[tex]\[ CI_B = \left( 0.82 - 1.96 \times 0.009604686356149274, 0.82 + 1.96 \times 0.009604686356149274 \right) \][/tex]
[tex]\[ CI_B = (0.8011748147419474, 0.8388251852580525) \][/tex]

Step 3: Determine the width of the confidence intervals.
The width of a confidence interval is given by the difference between the upper and lower bounds of the interval.

For Survey A:
[tex]\[ \text{CI width}_A = 0.900141324683347 - 0.859858675316653 = 0.0402826493666939 \][/tex]

For Survey B:
[tex]\[ \text{CI width}_B = 0.8388251852580525 - 0.8011748147419474 = 0.03765037051610509 \][/tex]

Step 4: Comparison of confidence intervals.

- The width of the confidence interval is smaller for Survey B: [tex]\( 0.03765037051610509 \)[/tex] compared to Survey A: [tex]\( 0.0402826493666939 \)[/tex].

Based on this analysis, the true statement is:
- The confidence interval is smaller for Survey B.