An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the width of the 90% confidence interval?
Question
An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be 15,000. What is the width of the 90% confidence interval?
Solution
To calculate the width of the 90% confidence interval, we first need to find the standard error. The standard error (SE) is calculated by dividing the standard deviation (SD) by the square root of the sample size (n).
In this case, the SD is $1,000 and the sample size is 50. So, the SE = SD / sqrt(n) = 1000 / sqrt(50) = 141.42.
Next, we need to find the z-score that corresponds to a 90% confidence interval. The z-score for a 90% confidence interval is 1.645 (you can find this value in a standard z-table or use a calculator that provides this value).
The margin of error (ME) is then calculated by multiplying the z-score by the SE. So, ME = z * SE = 1.645 * 141.42 = 232.54.
The width of the confidence interval is twice the margin of error. So, the width = 2 * ME = 2 * 232.54 = $465.08.
So, the width of the 90% confidence interval for the average income in this region is $465.08.
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