Asked by Charmaine Butler on Jun 03, 2024
Verified
For a variable with 
= 25.00, how large should the sample be for a desired 95% confidence interval width of 15?
A) 7
B) 11
C) 43
D) 121
Confidence Interval
A range of values, derived from the sample data, that is likely to contain the true population parameter with a specified level of confidence.
Sample Size
Sample size refers to the number of individual observations or data points collected and used in a statistical analysis.
Variable
Any characteristic, number, or quantity that can be measured or counted and which can vary across subjects in a study.
- Absorb the significance of sample size in affecting the resolution of confidence intervals.
Verified Answer
KS
Kevin SanabriaJun 06, 2024
Final Answer :
C
Explanation :
The sample size needed for a 95% confidence interval with a width of 15 is given by:
n = (Zα/2 * σ / E)^2
where Zα/2 is the critical value of the standard normal distribution for 95% confidence (which is 1.96), σ is the population standard deviation (which is unknown, so we will assume a conservative estimate of 10), and E is the margin of error, which is half the desired width of the confidence interval (so E = 15/2 = 7.5).
Plugging in these values, we get:
n = (1.96 * 10 / 7.5)^2
n = 42.95
Since we cannot have a fractional sample size, we round up to the nearest integer to get a sample size of 43. Therefore, the best answer choice is C.
n = (Zα/2 * σ / E)^2
where Zα/2 is the critical value of the standard normal distribution for 95% confidence (which is 1.96), σ is the population standard deviation (which is unknown, so we will assume a conservative estimate of 10), and E is the margin of error, which is half the desired width of the confidence interval (so E = 15/2 = 7.5).
Plugging in these values, we get:
n = (1.96 * 10 / 7.5)^2
n = 42.95
Since we cannot have a fractional sample size, we round up to the nearest integer to get a sample size of 43. Therefore, the best answer choice is C.
Learning Objectives
- Absorb the significance of sample size in affecting the resolution of confidence intervals.