Participants in independent representative samples of teens and parents of teens were asked if they think that they spend too much time on their cell phones. Of the 750 teens surveyed, 56% said "yes" and of the 1,050 parents of teens surveyed, 34% said "yes."
(a) Construct a 95% large-sample confidence interval for the difference in the proportion of teens who think they spend too much time on their cell phones and this proportion for parents of teens
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To construct a confidence interval for the difference in proportions, we can use the formula:
CI = (p1 - p2) ± z * sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
Where:
- p1 = proportion of teens who think they spend too much time on their cell phones
- p2 = proportion of parents of teens who think they spend too much time on their cell phones
- n1 = sample size of teens
- n2 = sample size of parents
- z = z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of 1.96)
Given:
- p1 = 56% = 0.56
- p2 = 34% = 0.34
- n1 = 750
- n2 = 1050
- z = 1.96
Plugging in these values into the formula, we have:
CI = (0.56 - 0.34) ± 1.96 * sqrt((0.56 * (1 - 0.56) / 750) + (0.34 * (1 - 0.34) / 1050))
Calculating the standard deviation within the square root:
sqrt((0.56 * (1 - 0.56) / 750) + (0.34 * (1 - 0.34) / 1050)) = 0.0161
Substituting this back into the formula, we have:
CI = 0.22 ± 1.96 * 0.0161
Calculating the bounds of the confidence interval:
CI = 0.22 ± 0.0316
This gives us the confidence interval (rounded to four decimal places):
CI = (0.1884, 0.2516)
Therefore, we can be 95% confident that the true difference in proportions for teens and parents of teens who think they spend too much time on their cell phones falls within the range of 0.1884 to 0.2516.
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