Study Guide for Chapter 9: Testing a Claim - Section 9.3

Chapter 9: Testing a Claim

Section 9.3: Tests About a Population Mean

Education Authors and Resources

• Source: Starnes/Tabor, The Practice of Statistics

• Publisher: Bedford, Freeman, & Worth High School Publishers

Learning Targets

By the end of this section, you should be able to:

• STATE and CHECK the Random, 10%, and Normal/Large Sample conditions for performing a significance test about a population mean.

• CALCULATE the standardized test statistic and P-value for a test about a population mean.

• PERFORM a significance test about a population mean.

• USE a confidence interval to MAKE a conclusion for a two-sided test about a population parameter.

• INTERPRET the power of a significance test and DESCRIBE what factors affect the power of a test.

Carrying Out a Significance Test for µ

Conditions for Validity

• Random Condition:

  • The data must come from a random sample from the population of interest.

• 10% Condition:

  • When sampling is done without replacement, the sample size must satisfy:

  • n < 0.10N where N is the population size.

• Normal/Large Sample Condition:

  • The population distribution must either be Normal or the sample size must be large $(n \geq 30)$.

  • If the population distribution is of unknown shape and $n < 30$, assess Normality using a graph of the sample data.

  • Do not apply t procedures if strong skewness or outliers are present in the graph.

• AP® Exam Tip:

  • It is essential to sketch the graph and make comments for scoring. A single appropriate graph suffices.

Example: Battery Lifetimes

• Problem: Given the lifetimes (in hours) of 15 deluxe AAA batteries from a simple random sample, check if conditions for performing the significance test are met.

• Analysis:

  • Random: SRS of 15 deluxe AAA batteries. Confirmed.

  • 10% Condition: Assumed that 15 is less than 10% of the total deluxe AAA batteries produced.

  • Normal/Large Sample: A dotplot analysis shows no strong skewness or outliers. Confirmed.

Standardized Test Statistic

Formula

• The formula for the standardized test statistic is:

  • $\text{standardized test statistic} = \frac{\text{statistic} - \text{parameter}}{\text{standard deviation of the statistic}}$

Standard Normal Distribution Reference

• For a hypothesis $H<em>0: \mu = \mu</em>0$, ideally we would use:

  • $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• Since the population standard deviation $\sigma$ is typically unknown, replace it with the sample standard deviation $s_x$, leading to:

  • $t = \frac{\bar{x} - \mu<em>0}{s</em>x / \sqrt{n}}$

Degrees of Freedom

• When performing inference about population means using a t distribution, use:

  • $df = n - 1$

Characteristics of t Distributions

• The t distributions are:

  • Symmetric about 0, single-peaked, bell-shaped.

  • More variable than the standard Normal distribution with increased tails.

  • As degrees of freedom increase, the t distributions increasingly resemble the standard Normal distribution.

Finding P-values with Table B

• Utilize Table B to find P-values from the appropriate t distribution.

• Example: For $H<em>0: \mu = 30$ and Ha: \mu > 30:

  • Calculated t statistic:

  • $t = \frac{33.93 - 30}{9.82 / \sqrt{15}} = 1.55$

  • Degrees of freedom:

  • $df = 15 - 1 = 14$

• Look up P(t ≥ 1.55) between 0.10 and 0.05.

Limitations of Table B

• It only provides positive t value probabilities.

• For negative t values, the symmetry of t distributions can be applied.

• Includes probabilities from $df = 1$ to $30$, then skips to higher degrees.

Performing a Hypothesis Test

Example Problem

• Test: $H<em>0: \mu = 5$ versus $H</em>a: \mu ≠ 5$ at $\alpha = 0.01$ with:

  • Sample size $n = 37$

  • Sample mean $\bar{x} = 4.81$

  • Sample standard deviation $s_x = 0.365$

Step (a): Evidence Explanation

• Sample mean of $\bar{x} = 4.81$ indicates evidence for $H_a$ implying that it is not equal to 5.

Step (b): Calculate Test Statistic and P-value

• $t = \frac{4.81 - 5}{0.365 / \sqrt{37}} = -3.17$

• Degrees of freedom: $df = 37 - 1 = 36$.

Using Table B

• Notably not available for df = 36, thus use df = 30.

• Identify:

  • $P(t ≥ 3.17)$ yields P-values between 0.001 and 0.0025; for two-sided, double it yielding ranges between 0.002 and 0.005.

Recommendations for P-values

• Given limitations of Table B, employ technology to calculate P-values accurately during hypothesis tests.

Four-Step Significance Testing Process

Steps Defined

1. State: Define hypotheses and significance levels, and denote parameters.

2. Plan: Identify the suitable inference method and check conditions.

3. Do: If conditions are satisfied:

   • Calculate the sample statistic(s).

   • Compute the standardized test statistic.

   • Determine the P-value.

4. Conclude: Derive conclusions regarding the hypotheses within the context of the problem.

Validity of the Standardized Test Statistic Formula

Necessary Conditions

1. Random Condition: Ensures $\bar{x} - \mu_0$ is a solid estimate for the true mean.

2. Normal/Large Sample Condition: Ensures distribution of $t$ can be modeled adequately using a t distribution with degrees of freedom being $n - 1$.

3. 10% Condition: Required when sampling without replacement from a finite population to maintain proper standard deviation formula for $\bar{x}$.

One-Sample t Test Process

• If validated conditions apply to test $H<em>0: \mu = \mu</em>0$, compute:

  • $t = \frac{\bar{x} - \mu<em>0}{s</em>x / \sqrt{n}}$

• Derive P-value through determining the probability associated with the specified $t$ statistic given degrees of freedom $n - 1$.

Example: Dissolved Oxygen Levels

• Research averaged DO at randomly selected stream locations, hypothesizing implications for aquatic life based on significant levels established:

  • Set $H<em>0: \mu = 5$ and Ha: \mu < 5, at $\alpha = 0.05$.

• Review conditions satisfaction (random sample, normal/large, appropriate sample size).

• Calculate $t$ value and corresponding P-value using appropriate methods.

Confidence Intervals Linking to Two-Sided Tests

• Two-sided tests can derive conclusions directly through confidence intervals.

• If a 95% confidence interval for $\mu$ does not include the null hypothesis value $\mu<em>0$, one can reject $H</em>0$ at the 0.05 significance level.

  • Alternatively, if the interval does include $\mu<em>0$, fail to reject $H</em>0$.

Significance versus Practical Importance

• Understanding statistical significance compared to practical importance is essential.

• Confidence intervals should accompany tests to evaluate practical significance.

• Data dredging/P-hacking are practices to be mindful of, as they undermine legitimate statistical analysis.

Closing Remarks on Learning Targets

Ensure capability to:

• State and check all relevant conditions.

• Calculate standardized test statistics and P-values.

• Perform tests appropriately under controlled parameters.

• Utilize confidence intervals effectively for hypothesis testing conclusions.

• Interpret the power of tests and outline its influencing factors.