An IQ test was administered to a random sample of 26 mental patients and the mean was found to be 99.7. For this test, the population values are known to be normally distributed and with a population standard deviation of 15. Construct a 90% confidence interval for the mean IQ of the population.
http://davidmlane.com/hyperstat/z_table.html
To construct a confidence interval for the mean IQ of the population, we can use the formula:
CI = X̄ ± Z * (σ/√n)
Where:
CI = Confidence Interval
X̄ = Sample Mean
Z = Z-Score (determined by the confidence level)
σ = Population Standard Deviation
n = Sample Size
In this case:
Sample Mean (X̄) = 99.7
Population Standard Deviation (σ) = 15
Sample Size (n) = 26
Confidence Level = 90% (which corresponds to a Z-Score)
Step 1: Find the Z-Score
To determine the Z-Score corresponding to a 90% confidence level, we need to calculate the critical value. Since the data is normally distributed, we can use standard normal distribution tables or a statistical software to find the Z-Score. For a 90% confidence level, the Z-Score is approximately 1.645.
Step 2: Calculate the Margin of Error
The margin of error represents the range around the sample mean within which the population mean is likely to fall. It is calculated by multiplying the Z-Score by the standard error.
Margin of Error = Z * (σ/√n)
Margin of Error = 1.645 * (15/√26)
Step 3: Calculate the Confidence Interval
The confidence interval is constructed by adding and subtracting the margin of error from the sample mean.
Confidence Interval = X̄ ± Margin of Error
Confidence Interval = 99.7 ± (1.645 * (15/√26))
Now we can solve the equation to find the lower and upper bounds of the confidence interval.