A simple random sample with n =54 provided

Grecia Cordova Ramirez

Grecia Cordova Ramirez

Answered question

2022-09-25

A simple random sample with n =54 provided a sample mean of  24.0 and a sample standard deviation of 4.1.

a. Develop a 90% confidence interval for the population mean (to 1 decimal).

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b. Develop a 95% confidence interval for the population mean (to 1 decimal).

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c. Develop a 99% confidence interval for the population mean (to 1 decimal).

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d. What happens to the margin of error and the confidence interval as the confidence level is increased?

Answer & Explanation

Jazz Frenia

Jazz Frenia

Skilled2023-05-29Added 106 answers

a. To develop a 90% confidence interval for the population mean, we can use the formula:
(x¯zsn,x¯+zsn)
where:
x¯ is the sample mean,
s is the sample standard deviation,
n is the sample size,
z is the critical value corresponding to the desired confidence level.
Given:
x¯=24.0,
s=4.1,
n=54.
To find the critical value, we can use a t-distribution table or a statistical software. For a 90% confidence level with 53 degrees of freedom, the critical value is approximately 1.675.
Plugging in the values, we get:
(24.01.6754.154,24.0+1.6754.154)
Simplifying the expression, we find:
(23.0,25.0)
Therefore, the 90% confidence interval for the population mean is (23.0,25.0).
b. Using the same approach, for a 95% confidence level with 53 degrees of freedom, the critical value is approximately 2.009. Plugging in the values, we find:
(24.02.0094.154,24.0+2.0094.154)
Simplifying the expression, we find:
(22.8,25.2)
Therefore, the 95% confidence interval for the population mean is (22.8,25.2).
c. For a 99% confidence level with 53 degrees of freedom, the critical value is approximately 2.684. Plugging in the values, we find:
(24.02.6844.154,24.0+2.6844.154)
Simplifying the expression, we find:
(22.2,25.8)
Therefore, the 99% confidence interval for the population mean is (22.2,25.8).
d. As the confidence level is increased, the margin of error and the width of the confidence interval increase. This means that we become more confident in capturing the true population mean, but the interval becomes wider, indicating greater uncertainty. The trade-off between confidence and precision is observed in confidence intervals. Higher confidence levels require larger intervals to accommodate the increased certainty, resulting in a wider range of possible population means.

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