Construct a 90% confidence interval estimate for the population mean given the following values:
The Correct Answer and Explanation is:
To construct a 90% confidence interval estimate for the population mean, we use the formula for a confidence interval:Confidence Interval=xˉ±Z×sn\text{Confidence Interval} = \bar{x} \pm Z \times \frac{s}{\sqrt{n}}Confidence Interval=xˉ±Z×ns
Where:
- xˉ\bar{x}xˉ = sample mean
- ZZZ = Z-score corresponding to the desired confidence level (for 90%, the Z-value is 1.645)
- sss = sample standard deviation
- nnn = sample size
Steps:
- Identify the sample mean (xˉ\bar{x}xˉ): This is the average of your sample data points.
- Identify the sample standard deviation (sss): This measures the spread of your data points around the sample mean.
- Identify the sample size (nnn): The number of data points in your sample.
- Z-value for a 90% confidence interval: The Z-score for a 90% confidence level is 1.645 (this corresponds to 5% in each tail of the standard normal distribution).
- Calculate the standard error: This is calculated as:
SE=snSE = \frac{s}{\sqrt{n}}SE=ns
- Calculate the margin of error: Multiply the standard error by the Z-value.
Margin of Error=Z×SE\text{Margin of Error} = Z \times SEMargin of Error=Z×SE
- Construct the confidence interval: Finally, the confidence interval is:
xˉ±Margin of Error\bar{x} \pm \text{Margin of Error}xˉ±Margin of Error
Example:
Suppose you are given the following values:
- Sample mean (xˉ\bar{x}xˉ) = 50
- Sample standard deviation (sss) = 10
- Sample size (nnn) = 25
Using the formula:SE=1025=105=2SE = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2SE=2510=510=2
The margin of error:Margin of Error=1.645×2=3.29\text{Margin of Error} = 1.645 \times 2 = 3.29Margin of Error=1.645×2=3.29
The confidence interval will be:50±3.2950 \pm 3.2950±3.29
So, the confidence interval is:[46.71,53.29][46.71, 53.29][46.71,53.29]
Thus, we are 90% confident that the population mean lies between 46.71 and 53.29.
This confidence interval provides a range of values within which we expect the true population mean to fall, with a 90% level of confidence.
