7. A recent study investigated the effects of a "Buckle Up Your Toddlers" campaign to get parents to use the grocery cart seat belts. When gathering baseline data prior to the campaign, investigators observed 86 out of 640 parents buckling up their toddlers. Assume it is reasonable to regard this sample as representative of the population of parents with toddlers.
a) Compute a point estimate of the true proportion of all parents who buckle up their toddlers. (2 pts)
b) Construct and interpret a 92% confidence interval for , the true proportion of all parents who buckle up their toddlers. (6 pts)
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a) To compute a point estimate of the true proportion of all parents who buckle up their toddlers, we need to calculate the sample proportion. The sample proportion is obtained by dividing the number of parents who buckle up their toddlers by the total number of parents observed.
In this case, the number of parents who buckle up their toddlers is 86, and the total number of parents observed is 640.
The formula to calculate the sample proportion is:
Sample Proportion (p̂) = Number of parents who buckle up their toddlers / Total number of parents observed
So, in this case, the point estimate of the true proportion of all parents who buckle up their toddlers is:
p̂ = 86 / 640 = 0.1344
Therefore, the point estimate of the true proportion is approximately 0.1344 or 13.44%.
b) To construct a confidence interval for the true proportion of all parents who buckle up their toddlers, we can use the formula:
Confidence Interval = p̂ ± Z * √(p̂ * (1 - p̂) / n)
Where:
p̂ is the sample proportion
Z is the Z-score corresponding to the desired confidence level
√ represents the square root
n is the total sample size
In this case, we want to construct a 92% confidence interval. The Z-score corresponding to a 92% confidence level is approximately 1.75.
So, the confidence interval is:
Confidence Interval = 0.1344 ± 1.75 * √(0.1344 * (1 - 0.1344) / 640)
Calculating the values inside the square root:
√(0.1344 * (1 - 0.1344) / 640) ≈ 0.0147
Plugging the values into the confidence interval formula:
Confidence Interval = 0.1344 ± 1.75 * 0.0147
Calculating the confidence interval:
Lower Limit = 0.1344 - (1.75 * 0.0147) ≈ 0.1069
Upper Limit = 0.1344 + (1.75 * 0.0147) ≈ 0.1619
Therefore, the 92% confidence interval for the true proportion of all parents who buckle up their toddlers is approximately 0.1069 to 0.1619, or 10.69% to 16.19%.
Interpretation: We are 92% confident that the true proportion of all parents who buckle up their toddlers is between 10.69% and 16.19%.