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Answer :
To find the proportion of U.S. adults who contribute to crowdfunding campaigns, first, we look at the given data:
- Total sample size (n): 1535 U.S. adults
- Number of adults who contribute (X): 487
The proportion ([tex]\hat{p}[/tex]) of U.S. adults in the sample who contribute to crowdfunding campaigns is calculated using the formula:
[tex]\hat{p} = \frac{X}{n} = \frac{487}{1535}[/tex]
Calculating this gives:
[tex]\hat{p} \approx 0.3172[/tex]
To express this proportion as a percentage, multiply by 100:
[tex]0.3172 \times 100 = 31.72\%[/tex]
Rounding this to the nearest whole number, approximately 32% of U.S. adults in this sample contribute to crowdfunding campaigns.
Next, we'll check if the large counts condition is met. The large counts condition requires that both [tex]n \times \hat{p}[/tex] and [tex]n \times (1 - \hat{p})[/tex] are at least 10.
Calculate [tex]n \times \hat{p}[/tex]:
[tex]1535 \times 0.3172 \approx 487.016[/tex]This is greater than 10.
Calculate [tex]n \times (1 - \hat{p})[/tex]:
[tex]1535 \times (1 - 0.3172) = 1535 \times 0.6828 \approx 1047.984[/tex]This is also greater than 10.
Since both numbers are greater than 10, the large counts condition is satisfied. Therefore, it's appropriate to use the normal approximation to model the sampling distribution of the sample proportion. Thus, about 32% of U.S. adults, according to this sample, typically contribute to crowdfunding campaigns like GoFundMe.
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