Estimating a Population Mean

Point Estimation and Confidence Intervals

In statistics, estimating the unknown value of a population mean (\(\mu\)) is a fundamental task. Often, the population standard deviation (\(\sigma\)) is unknown, so we rely on sample statistics. The sample mean (\(\bar{x}\)) is an unbiased estimator of \(\mu\), and it is considered the best point estimate due to its tendency to vary less than other measures of center.

• Point Estimate: The sample mean \(\bar{x}\) is used as the best estimate for the population mean \(\mu\).

• Confidence Interval: Provides a range of values likely to contain the population mean, offering a measure of accuracy for the estimate.

Student t Distribution

When \(\sigma\) is unknown and the sample size is small, the Student t distribution is used. The t-statistic is calculated as:

• Degrees of Freedom (df): The number of values in the sample that are free to vary, typically \(n - 1\).

• Critical Value (t\alpha/2): The value separating an area of \(\alpha/2\) in the right tail of the t distribution.

Figure 7-4: The Student t distribution has a symmetric bell shape similar to the standard normal distribution but with greater variability, especially for small samples.

Properties of the Student t Distribution

• The distribution varies with sample size.

• It is symmetric and bell-shaped, like the normal distribution.

• Mean is 0; standard deviation is greater than 1 and depends on sample size.

• As sample size increases, the t distribution approaches the normal distribution.

Constructing Confidence Intervals for the Population Mean

Requirements

• The sample must be a simple random sample.

• The population must be normally distributed or the sample size must be greater than 30 (\(n > 30\)).

Confidence Interval Formula

The confidence interval for \(\mu\) when \(\sigma\) is unknown is:

• Margin of Error (E):

• Confidence Level: The probability that the interval contains \(\mu\), e.g., 95% or 99%.

• Critical Value: is determined by the confidence level and degrees of freedom.

• Degrees of Freedom:

Finding Point Estimate and Margin of Error from Interval

• Point Estimate:

• Margin of Error:

Example: Old Faithful Geyser Eruption Times

A 95% confidence interval for eruption times is (85.74, 91.76) minutes. The sample mean is 88.75, sample standard deviation is 8.90, and sample size is 36.

• Interpretation: We have 95% confidence that the true mean eruption time is between 85.74 and 91.76 minutes.

• Requirements Met: The sample is random and \(n = 36\) (> 30).

Constructing a (1 - \(\alpha\)) Confidence Interval for \(\mu\) (\(\sigma\) Unknown)

For a simple random sample from a population with unknown mean and standard deviation, the confidence interval is:

where is based on degrees of freedom.

Example: Hershey Kisses Weights

• Sample size: 32

• Sample mean: 4.5210 g

• Sample standard deviation: 0.1077 g

• Confidence level: 99%

• Interpretation: We have 99% confidence that the true mean weight is between 4.4688 g and 4.5732 g.

Example: Garlic Treatment for Cholesterol

• Sample size: 49

• Mean change: 0.4 mg/dL

• Standard deviation: 21.0 mg/dL

• Confidence level: 98%

• Interpretation: The confidence interval includes 0, suggesting garlic may not significantly affect LDL cholesterol.

Example: Arsenic in Brown Rice

• Sample from California: 10 values

• Construct a 90% confidence interval for mean arsenic content.

• Note: The interval cannot be used for samples from Arkansas due to potential population differences.

Determining Sample Size

Sample Size Formula

To estimate the population mean with a specified margin of error (E) and confidence level, use:

• Round-Off Rule: If n is not a whole number, round up to the next whole number.

Estimating Standard Deviation: Range Rule of Thumb

If \(\sigma\) is unknown, estimate it using:

• Range is the difference between the highest and lowest sample values.

Example: Pulse Rates of Adult Males

• Sample size: 153

• Pulse rates range from 40 bpm to 104 bpm.

• 99% confidence, margin of error: 2 bpm.

• Estimate \(\sigma\) using range rule and compare with sample standard deviation.

• Conclusion: Using the sample standard deviation (s) yields a more accurate sample size than the range rule estimate.

Table: Sample data for pulse rate estimation, including age, gender, pulse, and other health metrics.