Estimating a Population Mean
In addition to estimating population proportions, inferential statistics also focuses on estimating population means. A population mean (\( \mu \)) is a central measure of a quantitative dataset, often unknown and estimated using sample data.
The Best Point Estimate for a Population Mean
Definition: Best Point Estimate for the Population Mean
The best point estimate for a population mean (\( \mu \)) is the sample mean (\( \bar{x} \)). The sample mean provides the most unbiased estimate of the population mean based on sample data. This means that sample means (\( \bar{x} \)) target the true population mean (\( \mu \)) as more samples are collected.
Example : Calculating the Best Point Estimate
A study collected pulse rates (beats per minute) from 10 randomly selected adults: 72, 75, 68, 80, 77, 74, 70, 73, 78, and 76. What is the best point estimate for the population mean pulse rate?
Solution
To find the sample mean, calculate the sum of all pulse rates and divide by the total number of observations:
\[ \bar{x} = \frac{72 + 75 + 68 + 80 + 77 + 74 + 70 + 73 + 78 + 76}{10} = 74.3 \]
Thus, the best point estimate for the population mean is 74.3 beats per minute.
Confidence Intervals for Estimating the Population Mean
Confidence intervals for population means provide a range of values within which the true mean is likely to lie. These intervals are calculated using the sample mean (\( \bar{x} \)), sample standard deviation (\( s \)), and the sample size (\( n \)) as well as technology (such as GeoGebra) to find the upper and lower bounds.
Interactive Confidence Interval Tool
Interactive calculator for confidence intervals and hypothesis testing.
Opens the calculator in a new tab.
Definition: Confidence Interval Representations for Estimating Population Mean
A confidence interval estimate for the population mean can be represented in three different ways
\[(\text{lower bound} , \text{upper bound})\]
\[\text{lower bound} < \mu < \text{upper bound}\]
\[\bar{x} \pm E \]
Where:
- \( \bar{x} \): Sample mean
- \( E \): Margin of Error
Note: Margin of Error \(E\) and sample mean \(\bar{x}\) can be found with the following formulas if you are given the confidence interval:
- \( \bar{x} = \cfrac{\text{upper bound}+\text{lower bound}}{2} \)
- \( E = \cfrac{\text{upper bound}-\text{lower bound}}{2}\)
Examples: Confidence Intervals for a Mean
Example : Confidence Interval for Mean Study Time
A random sample of 25 students recorded an average study time of 6.4 hours per week with a standard deviation of 1.2 hours. Use this information to answer the following questions:
- Find the best point estimate for the mean study time.
- Find a 95% confidence interval estimate for the mean study time.
- Interpret the confidence interval found in part (b).
- Would it be reasonable to claim that the true population mean study time is 7.5 hours per week? Use the confidence interval from part (b) to justify the answer.
Solution
Part (a): The best point estimate for the mean study time is the sample mean, which is:
\( \bar{x} = 6.4 \) hours
Part (b): Using the GeoGebra Confidence Interval and Hypothesis Test Calculator, the 95% confidence interval for the mean study time is:
\( 5.905 , 6.895 \) hours (rounded to three decimal places)
Part (c): Interpretation:
We are 95% confident that the interval from 5.9 hours to 6.9 hours actually contains the true population mean time for students studying in a week.
Part (d): Since 7.5 hours is not within the confidence interval from part (b), it would not be reasonable to claim that the true population mean study time is 7.5 hours per week. The interval suggests that the true mean is likely between 5.9 and 6.9 hours.
Example : Pulse Rate Data
A study measured the resting pulse rates (in beats per minute) of 30 males and 30 females. Use the pulse rate dataset linked below to construct a 95% confidence interval for the mean pulse rate for males and then for females. Write the confidence intervals for each in Margin of Error format \( \bar{x} \pm E \).Then compare the two intervals to determine if there is evidence of a significant difference between the two groups.
Because of the amount of data, this dataset is omitted from viewing but can be viewed visually in a scrollable table by clicking the button below or by downloading the data here: Download Pulse Rate Data.
This button loads and displays data from a CSV file with pulse rate information on this page.
Solution
1. For males:
- \( \bar{x}_{male} = 74.5 \)
- \( s_{male} = 2.1 \)
- \( n_{male} = 30 \)
- \( t^* = 2.045 \) (from the \( t \)-distribution table for 29 degrees of freedom at 95% confidence)
The confidence interval is:
\[ \bar{x}_{male} \pm t^* \frac{s_{male}}{\sqrt{n_{male}}} = 74.5 \pm 2.045 \cdot \frac{2.1}{\sqrt{30}} = 74.5 \pm 0.785 \]
Confidence interval for males: \([73.715, 75.285]\).
2. For females:
- \( \bar{x}_{female} = 81.5 \)
- \( s_{female} = 1.8 \)
- \( n_{female} = 30 \)
- \( t^* = 2.045 \) (from the \( t \)-distribution table for 29 degrees of freedom at 95% confidence)
The confidence interval is:
\[ \bar{x}_{female} \pm t^* \frac{s_{female}}{\sqrt{n_{female}}} = 81.5 \pm 2.045 \cdot \frac{1.8}{\sqrt{30}} = 81.5 \pm 0.673 \]
Confidence interval for females: \([80.827, 82.173]\).
The confidence interval for males in margin of error format is:
\( 74.5 \pm 0.785 \)
The confidence interval for females in margin of error format is:
\( 81.5 \pm 0.673 \)
The male and female intervals do not overlap, indicating a statistically significant difference in the mean pulse rates between the two groups. This suggests that males and females may have different resting pulse rates on average.