A psychologist is studying the self image of smokers, which she measures by the self-image (SI) score from a personality inventory. She would like to estimate the mean SI score, μ, for the population of all smokers. She plans to take a random sample of SI scores for smokers and estimate μ via this sample. Assuming that the standard deviation of SI scores for the population of all smokers is 90, what is the minimum sample size needed for the psychologist to be 95% confident that her estimate is within 15 of μ?Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).(If necessary, consult a list of formulas.)
Question
A psychologist is studying the self image of smokers, which she measures by the self-image (SI) score from a personality inventory. She would like to estimate the mean SI score, μ, for the population of all smokers. She plans to take a random sample of SI scores for smokers and estimate μ via this sample. Assuming that the standard deviation of SI scores for the population of all smokers is 90, what is the minimum sample size needed for the psychologist to be 95% confident that her estimate is within 15 of μ?Carry your intermediate computations to at least three decimal places. Write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).(If necessary, consult a list of formulas.)
Solution
To solve this problem, we need to use the formula for the sample size in a confidence interval estimation, which is:
n = (Z*σ/E)^2
where:
- n is the sample size
- Z is the Z-score (which corresponds to the desired confidence level, in this case 95%)
- σ is the standard deviation of the population
- E is the margin of error (the amount we want to be within the true mean)
For a 95% confidence level, the Z-score is approximately 1.96 (you can find this value in a standard Z-table or use a calculator that provides this value).
Given that σ = 90 and E = 15, we can substitute these values into the formula:
n = (1.96*90/15)^2
First, calculate the value inside the parentheses:
1.96*90/15 = 11.76
Then, square this value to find n:
n = (11.76)^2 = 138.2976
Since we can't have a fraction of a person, we round up to the nearest whole number to ensure that the sample size is large enough to meet the requirements. Therefore, the minimum sample size needed is 139.
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