Central Limit Theorem, Confidence Intervals, and Hypothesis Testing: Study Notes

Central Limit Theorem (CLT) and Its Applications

Definition and Importance of the CLT

The Central Limit Theorem (CLT) is a fundamental result in statistics that describes the distribution of sample means. It states that, for a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, provided the population has a finite mean and variance.

• Population Mean (μ): The average value in the population.

• Population Standard Deviation (σ): The spread of values in the population.

• Sample Size (n): Number of observations in the sample.

• Sample Mean (\( \bar{x} \)): The average value in the sample.

Key Formula:

$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

This formula standardizes the sample mean to a standard normal distribution (mean 0, standard deviation 1).

Example: Probability of Sample Mean

• Scenario: The average weight of women is 140 lbs with a standard deviation of 10 lbs. Find the probability that the mean weight of a random sample of 25 women is between 139 and 141 lbs.

• Given: \( \mu = 140 \), \( \sigma = 10 \), \( n = 25 \)

• Standard Error: \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{10}{5} = 2 \)

• Calculate Z-scores:

$Z_1 = \frac{139 - 140}{2} = -0.5$ $Z_2 = \frac{141 - 140}{2} = 0.5$

• Using the standard normal table:

$P(-0.5 < Z < 0.5) = P(Z < 0.5) - P(Z < -0.5)$ $= 0.6915 - 0.3085 = 0.3830$ So, there is a 38.3% probability that the sample mean is between 139 and 141 lbs.

Confidence Intervals for the Mean

Constructing a Confidence Interval (CI)

A confidence interval provides a range of values within which the population parameter is likely to fall, with a certain level of confidence (e.g., 95%).

• Formula for CI when σ is known:

$\bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$

• \( z_{\alpha/2} \): Critical value from the standard normal distribution for the desired confidence level.

• \( \alpha \): Significance level (e.g., 0.05 for 95% confidence).

Example: Average Age of Students

• Scenario: What is the average age of all students today? A random sample of 225 students has a mean age of 20.4 years. Assume σ = 2.0 years.

• Given: \( \bar{x} = 20.4 \), \( \sigma = 2.0 \), \( n = 225 \)

• Standard Error: \( \frac{2.0}{\sqrt{225}} = 0.133 \)

• For 95% confidence, \( z_{0.025} = 1.96 \):

$CI = 20.4 \pm 1.96 \times 0.133 = 20.4 \pm 0.261 = (20.139, 20.661)$

We are 95% confident that the true mean age is between 20.139 and 20.661 years.

Hypothesis Testing for the Mean

Setting Up Hypotheses

Hypothesis testing is used to make inferences about population parameters. The null hypothesis (\( H_0 \)) is a statement of no effect or status quo, while the alternative hypothesis (\( H_a \)) is what we seek evidence for.

• Example Hypotheses:

$H_0: \mu \geq 20$ $H_a: \mu < 20$

If the sample mean is 20.4, it is not unreasonable to accept the null hypothesis at the 95% confidence level.

Standard Normal Distribution Table

Using the Z-Table

The standard normal distribution table (Z-table) provides the probability that a standard normal variable is less than a given value. It is used to find probabilities and critical values for hypothesis testing and confidence intervals.

Z	Probability
-0.5	0.3085
0.5	0.6915
Between -0.5 and 0.5	0.3830

Additional info: The Z-table values are used to calculate the probability that the sample mean falls within a certain range.