1.Leana is interested in the proportion of people who live in her area who own a dog. She randomly selects 50 addresses within a one-mile radius of her home and visits each to ask if they own at least one dog. Leana finds that 29 of the 50 homes she visited had at least one dog Construct a 98% confidence interval for the population proportion of people who live within one mile of Leana's home who own at least one dog.
To construct a confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
First, we need to calculate the sample proportion, which is the number of successes (29 homes with at least one dog) divided by the sample size (50 homes visited):
Sample Proportion (p̂) = number of successes / sample size
= 29/50
= 0.58
Next, we need to calculate the margin of error. The margin of error represents the maximum likely difference between the sample proportion and the population proportion.
The formula for the margin of error is:
Margin of Error = z * √((p̂ * (1 - p̂)) / n)
where:
- z is the z-value corresponding to the desired confidence level (in this case, 98% confidence level).
- p̂ is the sample proportion.
- n is the sample size.
Since the confidence level is 98%, the corresponding z-value can be found using a standard normal distribution table or calculator. The z-value for a 98% confidence level is approximately 2.33.
Now we can calculate the margin of error:
Margin of Error = 2.33 * √((0.58 * (1 - 0.58)) / 50)
= 2.33 * √(0.24 / 50)
≈ 0.163
Finally, we can construct the confidence interval by subtracting and adding the margin of error to the sample proportion:
Confidence Interval = Sample Proportion ± Margin of Error
= 0.58 ± 0.163
Therefore, the 98% confidence interval for the population proportion of people who live within one mile of Leana's home who own at least one dog is approximately 0.417 to 0.743.