Consider a 95% confidence interval for a population mean constructed from a random sample of 250 observations taken from a Normal population. Sigma is known to be 14. If we were to construct another 95% confidence interval, this time using 1,000 observations, how would the length of this interval compare to the length of the original interval?
half.
To compare the length of another 95% confidence interval using 1,000 observations to the length of the original interval, we need to understand how the length of a confidence interval is determined.
The length of a confidence interval is mainly influenced by three factors: the sample size (n), the standard deviation (σ), and the level of confidence (1 - α).
In this case, the original interval was constructed using a sample size (n) of 250 observations and a known standard deviation (σ) of 14. The level of confidence (1 - α) is 95%.
We can use the formula for the confidence interval for a population mean:
CI = X̄ ± z * σ/√n
Where:
CI = Confidence Interval
X̄ = Sample Mean
z = Z-score corresponding to the desired level of confidence
σ = Population Standard Deviation
n = Sample Size
In the original interval, since σ is known and n is 250, the length of the interval depends on the Z-score and σ/√n.
To construct another 95% confidence interval with 1,000 observations, the sample size (n) increases from 250 to 1,000. Assuming all other factors remain the same (including the same level of confidence and known population standard deviation), the length of the new interval would be shorter.
This is because a larger sample size (1,000) leads to a smaller value of √n in the denominator of the formula, which results in a narrower interval. The Z-score for a 95% confidence interval remains constant.
In conclusion, the length of the new interval with 1,000 observations would be smaller than the length of the original interval with 250 observations.