A magazine provided results from a poll of 500 adults who were asked to identify their favorite pie. Among the 500 respondents, [tex]14 \%[/tex] chose chocolate pie, and the margin of error was given as [tex]\pm 3[/tex] percentage points. What values do [tex]\hat{p}[/tex], [tex]\hat{q}[/tex], [tex]n[/tex], [tex]E[/tex], and [tex]p[/tex] represent?
If the confidence level is [tex]95 \%[/tex], what is the value of [tex]\alpha[/tex]?
Answer (1)
p ^ is the sample proportion: p ^ = 0.14 .
q ^ is calculated as 1 − p ^ : q ^ = 1 − 0.14 = 0.86 .
n is the sample size: n = 500 .
E is the margin of error: E = 0.03 , and α = 1 − 0.95 = 0.05 . The value of α is 0.05 .
Explanation
Analyze the problem and data We are given the results of a poll where 500 adults were asked to identify their favorite pie. We need to identify the values of p ^ , q ^ , n , E , and p , and also find the value of α given a 95% confidence level.
Identify the sample proportion The problem states that 14% of the 500 respondents chose chocolate pie. This is the sample proportion, so p ^ = 0.14 .
Calculate q ^ q ^ is calculated as 1 − p ^ . Therefore, q ^ = 1 − 0.14 = 0.86 .
Identify the sample size The sample size, n , is the number of adults polled, which is 500. So, n = 500 .
Identify the margin of error The margin of error, E , is given as ± 3 percentage points, which is 0.03 . So, E = 0.03 .
Determine what p represents p represents the sample proportion, which is the proportion of adults in the sample who chose chocolate pie.
Calculate alpha The confidence level is given as 95%, or 0.95. The value of α is calculated as 1 − confidence level . Therefore, α = 1 − 0.95 = 0.05 .
State the final answer In summary: p ^ = 0.14 q ^ = 0.86 n = 500 E = 0.03 p represents the sample proportion. α = 0.05
Examples
Understanding polls and surveys is crucial in many real-world applications. For instance, political campaigns use polls to gauge voter sentiment and adjust their strategies. Businesses use surveys to understand customer preferences and improve their products. The margin of error and confidence level help to quantify the uncertainty in these estimates, ensuring that decisions are based on reliable information. For example, if a poll shows that 55% of voters support a candidate with a margin of error of ± 3 %, it means the true support could be anywhere between 52% and 58%.
