Question

Critical values for quick reference during this activity. Confidence level/Critical value 0.90 z ∗ = 1.645 0.95 z ∗ = 1.960 0.99 z ∗ = 2.576 A hunger relief organization is trying to determine the lowest likely proportion of the local population that lives within two miles of a grocery store. A random sample of 50 local citizens was polled. Use the data below to calculate p ^ and the 95 % confidence interval. p ^ = Upper bound for 95 % confidence interval = Lower bound for 95 % confidence interval = The organization can say with 95 % confidence that the true population proportion of the local population that lives within two miles of a grocery store is in the interval [ ___,___ ] .The lowest likely proportion is ___ Within 2 miles No No No No No Yes Yes No Yes No No Yes No Yes Yes Yes Yes No No Yes No Yes Yes No Yes Yes No No No No Yes No Yes Yes No Yes Yes Yes No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes

Answer 1

The organization can state with 95% confidence that the true population proportion of the local population living within two miles of a grocery store is in the interval [0.616, 0.864]. The lowest likely proportion is 0.616.

Step-by-step explanation:

To calculate the proportion (p-hat) and the 95% confidence interval for the proportion of the local population living within two miles of a grocery store, we count the number of "Yes" responses in the sample.

Number of "Yes" responses = 37

Sample size (n) = 50

p-hat = Number of "Yes" responses / Sample size = 37 / 50 = 0.74

Using the critical value for a 95% confidence level (z* = 1.960):

Margin of Error = z* * sqrt(p-hat * (1 - p-hat) / n)

Margin of Error = 1.960 * sqrt(0.74 * (1 - 0.74) / 50)

Calculate the margin of error and apply it to find the lower and upper bounds of the 95% confidence interval:

Lower bound = p-hat - Margin of Error

Upper bound = p-hat + Margin of Error

Finally, the organization can state with 95% confidence that the true population proportion of the local population living within two miles of a grocery store is in the interval [0.616, 0.864]. The lowest likely proportion is 0.616.