Chapter 9

Chapter 9: Inferring Population Means

Learning Objectives

• Understand the distribution of the sample mean and why it is a good estimator of the population mean.

• Apply the Central Limit Theorem (CLT) for populations with Normal or any distribution.

• Compute and interpret confidence intervals for a population mean.

• Develop and conduct statistical hypothesis tests about the value of a population mean, distinguishing between hypothesis tests and confidence intervals.

Section 9.1: Sample Means of Random Samples

Three Characteristics of the Sample Mean

The sample mean (\( \bar{x} \)) is a random variable used to estimate the population mean (\( \mu \)). Its properties are:

• Accuracy: On average, the sample mean equals the population mean (unbiased estimator).

• Precision: The variability of the sample mean decreases as sample size increases.

• Probability Distribution: The sample mean has its own distribution, called the sampling distribution.

Notation Review

• \( \mu \): Population mean

• \( \bar{x} \): Sample mean

• \( \sigma \): Population standard deviation

• \( s \): Sample standard deviation

Example: Accuracy and Precision

• Accuracy: Sample mean is close to the population mean.

• Precision: Sample means are close to each other (low variability).

• Illustration: Target diagrams show combinations of accuracy and precision (e.g., shots clustered at the center are both accurate and precise).

Sampling Distribution of the Sample Mean

The sampling distribution is the probability distribution of all possible sample means from samples of a fixed size drawn from the population.

• As sample size increases, the sampling distribution becomes more concentrated around the population mean.

• The standard deviation of the sampling distribution (standard error) decreases as sample size increases.

Simulation Example

Sample	Q1	Q2	Q3	Q4	Q5	Average	Deviation from Mean
Sim#1	31	32	29	27	26	29.0	-1.3
Sim#2	36	35	34	32	31	33.6	3.3
...	Additional info: Table continues for other simulations, showing how sample means vary around the population mean.

Key Formulas

• Bias of the sample mean: $\text{Bias} = E(\bar{x}) - \mu$

• Standard deviation of the sample mean (Standard Error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Variance of the sample mean: $\text{Var}(\bar{x}) = \frac{\sigma^2}{n}$

Example: Calculating Sample Variance

• Population variance = 150, sample size = 30

• Sample variance = $\frac{150}{30} = 5$

Section 9.2: Central Limit Theorem for Sample Means

The Central Limit Theorem (CLT): Conditions

The CLT states that, under certain conditions, the sampling distribution of the sample mean is approximately Normal, regardless of the population's distribution.

• Random Sample: Observations are randomly selected and independent.

• Normality: Either the population is Normal, or the sample size is large (usually $n \geq 25$).

• Big Population: If sampling without replacement, the population should be at least 10 times larger than the sample size.

CLT: Conclusions

• The sampling distribution of $\bar{x}$ is approximately Normal.

• Mean of the sampling distribution: $\mu_{\bar{x}} = \mu$

• Standard deviation (standard error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Example: Estimating Probabilities with the CLT

• Given: Mean number of cigarettes = 5, SD = 6, sample size = 1000

• Standard error: $\frac{6}{\sqrt{1000}} = 0.19$

• Probability calculations use the Normal distribution with mean 5 and SD 0.19.

Visualizing the CLT

• As sample size increases, the sampling distribution becomes more Normal and less spread out.

• Even for non-Normal populations, the sample mean's distribution approaches Normality as $n$ increases.

Section 9.3: Answering Questions About the Mean of a Population

Two Approaches

• Confidence Intervals: Estimate parameter values.

• Hypothesis Tests: Decide whether a parameter's value is one thing or another.

Confidence Intervals for a Population Mean

A confidence interval gives a range of plausible values for the population mean, along with a confidence level (e.g., 95%).

• Conditions:

  • Random sample

  • Normality (population Normal or $n \geq 25$)

• Interpretation: The confidence level is the long-run proportion of intervals that will contain the population mean if the procedure is repeated many times.

Margin of Error

• Margin of error = (multiplier) × standard error

• For unknown $\sigma$, use $s$ and the $t$-distribution:

• $\text{Margin of error} = t^* \cdot \frac{s}{\sqrt{n}}$

Confidence Interval Formula

• $\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}$

• $t^*$ is the critical value from the $t$-distribution with $n-1$ degrees of freedom for the desired confidence level.

Example: Constructing a Confidence Interval

• Sample mean = 5218, $s$ = 893, $n$ = 34, 95% CI

• $t^* = 2.042$ (from $t$-table, df = 33)

• Standard error: $\frac{893}{\sqrt{34}} = 153.18$

• CI: $5218 \pm 2.042 \times 153.18 = (4906, 5530)$

Finding the Multiplier $t^*$

• Use $t$-tables or technology for the appropriate degrees of freedom and confidence level.

Interpretation of the Confidence Level

• We are 95% confident that the interval contains the true population mean.

• We cannot say there is a 95% probability that the specific interval contains the mean; the confidence level refers to the method, not a particular interval.

Examples: Additional Calculations

• Standard error for means: $SD_{est} = \frac{s}{\sqrt{n}}$

• t-critical values: e.g., $t_{29, 0.95} = 2.045$ for 95% CI with 29 df

• Margin of error: $ME = t^* \cdot SD_{est}$

Summary Table: Confidence Intervals

Source	Mean	SD	n	CI
Sample	52.11	8.32	31	48.95, 55.41
Additional info: Table shows how to compute CI for different samples.

Hypothesis Testing for a Population Mean

The Four Steps in Hypothesis Testing

1. Hypothesis: State null ($H_0$) and alternative ($H_A$) hypotheses about the population mean.

2. Prepare: Choose the test statistic, state/check conditions, select significance level ($\alpha$).

3. Compute & Compare: Calculate the test statistic and p-value; compare to $\alpha$.

4. Interpret & Conclude: State the result in context; never "accept" $H_0$, only fail to reject or reject.

Types of Hypotheses

Type	Null Hypothesis	Alternative Hypothesis
Two-Tailed	$H_0: \mu = \mu_0$	$H_A: \mu \neq \mu_0$
One-Tailed (Left)	$H_0: \mu = \mu_0$	$H_A: \mu < \mu_0$
One-Tailed (Right)	$H_0: \mu = \mu_0$	$H_A: \mu > \mu_0$

Test Statistics

• z-test: Use when population standard deviation $\sigma$ is known and CLT conditions are met. $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• t-test: Use when $\sigma$ is unknown; use sample standard deviation $s$ and $t$-distribution. $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Level of Significance ($\alpha$)

• Common values: 0.05, 0.01

• Represents the probability of rejecting $H_0$ when it is true (Type I error).

p-Values

• The probability, under $H_0$, of obtaining a result as extreme or more extreme than the observed sample statistic.

• If $p$-value < $\alpha$, reject $H_0$.

Example: Complete Hypothesis Test

• Given: $\mu_0 = 14.3$, sample mean $= 17.1$, $s = 12.2$, $n = 35$, $\alpha = 0.05$

• Calculate $t$:

• $t = \frac{17.1 - 14.3}{12.2 / \sqrt{35}} = 1.32$

• Find p-value (from $t$-table or technology); compare to $\alpha$.

• Interpret: If $p$-value < 0.05, conclude mean years of experience has increased.

Additional Practice and Concept Checks

• As sample size increases, the standard deviation of the sample mean decreases.

• For a population with $\mu = 20$, $\sigma = 35$, and $n = 100$, the standard deviation of the sample mean is $\frac{35}{10} = 3.5$.

• CLT applies for $n \geq 25$ for non-Normal populations.

• Choosing one-tailed or two-tailed tests depends on the research question and alternative hypothesis.