Carrying Out a Test for a Population Mean - AP Statistics Study Guide

Quick Summary

This guide focuses on the mechanics of conducting a significance test for a single population mean (μ). You will learn how to calculate the t-test statistic and its corresponding p-value. Mastering these calculations will allow you to use the p-value to make a statistically sound decision, justifying a claim about a population mean with a clear, evidence-based conclusion written in the context of the problem.

Key Concepts

The primary goal of a test for a population mean is to use sample data to assess the evidence against a claim about an unknown population mean, μ. Because the population standard deviation (σ) is almost always unknown, we use the sample standard deviation (sₓ) as an estimate. This requires us to use a t-distribution rather than a Normal distribution.

• The Test Statistic: The One-Sample t-statistic

  The test statistic measures how far the sample mean (x̄) deviates from the hypothesized population mean (μ₀), in units of standard error.

  • Formula:

    $t=(\bar{x}-{\mu }_0)/(s_x/\surd n)$

  • Where:

    • x̄ is the sample mean.

    • μ₀ is the hypothesized value of the population mean from the null hypothesis (H₀).

    • sₓ is the sample standard deviation.

    • n is the sample size.

• The t-distribution

  When we use the sample standard deviation (sₓ) to estimate the population standard deviation (σ), our test statistic follows a t-distribution.

  • Key Properties:

    • It is symmetric and bell-shaped, centered at 0, just like the standard Normal (z) distribution.

    • It has more variability (i.e., more area in the tails) than the standard Normal distribution. This accounts for the extra uncertainty introduced by estimating σ with sₓ.

    • As the sample size $n$ increases, the t-distribution gets closer to the standard Normal distribution.

  • Degrees of Freedom (df): A specific t-distribution is defined by its degrees of freedom. For a one-sample test for a population mean, the degrees of freedom are calculated as:

    • df = n - 1

  [Image: A graph comparing a standard Normal (z) curve with a t-distribution curve (e.g., with df=4). The t-distribution is slightly shorter and wider, showing more area in the tails.]

• Calculating the P-value

  The p-value is the probability of getting a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis (H₀) is true. The calculation depends on the alternative hypothesis (Hₐ).

  1. Right-Tailed Test (Hₐ: μ > μ₀): The p-value is the area to the right of your calculated t-statistic.

     • P-value = P(t \ge calculated t-statistic)

     [Image: A t-distribution curve with the area in the right tail shaded, representing the p-value.]

  2. Left-Tailed Test (Hₐ: μ < μ₀): The p-value is the area to the left of your calculated t-statistic.

     • P-value = P(t \le calculated t-statistic)

     [Image: A t-distribution curve with the area in the left tail shaded, representing the p-value.]

  3. Two-Tailed Test (Hₐ: μ \neq μ₀): The p-value is the area in both tails combined. You find the area in one tail and double it.

     • P-value = 2 * P(t \ge |calculated t-statistic|)

     [Image: A t-distribution curve with the areas in both the left and right tails shaded, representing the p-value.]

• Making a Conclusion

  The final step is to compare your p-value to the pre-determined significance level (α).

  • If p-value \le α: The result is statistically significant. We reject the null hypothesis (H₀). There is convincing evidence to support the alternative hypothesis (Hₐ).

  • If p-value > α: The result is not statistically significant. We fail to reject the null hypothesis (H₀). There is not convincing evidence to support the alternative hypothesis (Hₐ).

  • Your conclusion must always be stated in the context of the problem, clearly communicating the real-world implication of your decision.

Key Vocabulary

• Test Statistic: A value calculated from sample data that measures how far a sample statistic is from the parameter value stated in the null hypothesis. For a population mean, this is the t-statistic.

• t-distribution: A family of probability distributions that is used when the population standard deviation is unknown and is estimated from the sample data. It is defined by its degrees of freedom.

• Degrees of Freedom (df): For a one-sample t-test, it is the sample size minus one (n-1). It specifies which t-distribution to use for the analysis.

• P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample, assuming the null hypothesis is true. A small p-value provides evidence against the null hypothesis.

• Significance Level (α): A pre-determined threshold used to decide whether a p-value is small enough to reject the null hypothesis. Common values are 0.05 and 0.01.

Calculator Tech (TI-84)

The primary function for this topic is the $T-Test$, which calculates the test statistic and p-value for you.

Path:STAT -> TESTS -> 2: T-Test...

You will be prompted to choose an input method:

• $Data$: Use this if you have the raw data entered into a list (e.g., L1).

• $Stats$: Use this if you are given the summary statistics (sample mean, sample standard deviation, sample size).

Using the $Stats$ Input Method:

1. Select $Stats$ and press ENTER.

2. ${\mu }_0$: Enter the hypothesized population mean from your null hypothesis (H₀).

3. $\bar{x}$: Enter the sample mean.

4. $Sx$: Enter the sample standard deviation.

5. $n$: Enter the sample size.

6. $\mu$: Select the form of your alternative hypothesis (Hₐ). Choose $\ne {\mu }_0$ (two-tailed), $<{\mu }_0$ (left-tailed), or $>{\mu }_0$ (right-tailed).

7. Highlight $Calculate$ and press ENTER.

Output Screen:

The calculator will display:

• The alternative hypothesis you selected.

• $t=$: Your calculated t-test statistic.

• $p=$: Your calculated p-value.

• $\bar{x}$, $Sx$, and $n$ that you entered.

How to Show Work on the FRQ

To earn full credit on an inference question, you must clearly communicate every step of your reasoning. Use the four-step State-Plan-Do-Conclude framework.

STATE:

• Parameter: Define the parameter of interest in context. (e.g., "Let μ = the true mean weight of bags of chips produced at this factory.")

• Hypotheses: State the null (H₀) and alternative (Hₐ) hypotheses using symbols and context. (e.g., "H₀: μ = 10 ounces. Hₐ: μ \neq 10 ounces.")

• Significance Level: State the significance level, α. (e.g., "We will use a significance level of α = 0.05.")

PLAN:

• Procedure: Name the inference procedure. (e.g., "We will perform a one-sample t-test for a population mean.")

• Conditions: Check the three conditions required for this test.

  1. Random: State that the data come from a random sample or randomized experiment.

  2. 10% Condition: If sampling without replacement, check that the sample size $n$ is no more than 10% of the population size $N$ ($n\le 0.10N$).

  3. Normal/Large Sample: State that the population is approximately Normal, OR the sample size is large ($n\ge 30$), OR a graph of the sample data (like a boxplot or dotplot) shows no strong skewness or outliers.

DO:

• General Formula: Write the general formula for the test statistic. $TestStatistic=(Statistic-Parameter)/(StandardErrorofStatistic)$

• Specific Formula: Write the specific formula for the t-statistic. $t=(\bar{x}-{\mu }_0)/(s_x/\surd n)$

• Calculations: Plug the values from the problem into the formula and show the calculated test statistic.

  $t=(9.95-10)/(0.12/\surd 40)=-2.635$

• P-value: State the degrees of freedom ($df=n-1$). Report the p-value obtained from your calculator or a t-table.

  $Withdf=40-1=39,thep-valueisP(t\le -2.635)\ast 2\approx \mathrm{0.012.}$

CONCLUDE:

• Decision: Compare the p-value to α and make a decision about the null hypothesis.

  "Because our p-value of 0.012 is less than our significance level of α = 0.05, we reject the null hypothesis."

• Context: State your conclusion in the context of the original problem, referring to the alternative hypothesis.

  "We have convincing evidence that the true mean weight of the bags of chips produced at this factory is different from 10 ounces."

Practice Problems

Problem 1:

A coffee machine is designed to dispense an average of 12 fluid ounces of coffee per cup. A quality control manager is concerned the machine is under-filling the cups. They take a random sample of 20 cups and find a sample mean of 11.8 fluid ounces with a sample standard deviation of 0.4 fluid ounces. Is there convincing evidence at the α = 0.05 significance level that the machine is dispensing less than 12 ounces on average?

Solution:

STATE:

• Parameter: Let μ = the true mean amount of coffee dispensed by this machine in fluid ounces.

• Hypotheses:

  • H₀: μ = 12

  • Hₐ: μ < 12

• Significance Level: α = 0.05

PLAN:

• Procedure: We will perform a one-sample t-test for a population mean.

• Conditions:

  1. Random: The problem states a "random sample of 20 cups" was taken.

  2. 10% Condition: It is reasonable to assume that 20 cups is less than 10% of all cups of coffee this machine could dispense.

  3. Normal/Large Sample: The sample size (n=20) is not large (\ge 30). We must assume the distribution of coffee amounts is approximately Normal.

DO:

• Specific Formula:$t=(\bar{x}-{\mu }_0)/(s_x/\surd n)$

• Calculations:

  $t=(11.8-12)/(0.4/\surd 20)=-0.2/0.0894=-2.236$

• P-value:

  • Degrees of freedom: $df=n-1=20-1=19$.

  • The p-value is $P(t\le -2.236)$ with df=19. Using a calculator ($tcdf(-1E99,-2.236,19)$), we find the p-value is approximately 0.0188.

CONCLUDE:

• Decision: Because our p-value of 0.0188 is less than our significance level of α = 0.05, we reject the null hypothesis.

• Context: We have convincing evidence that the true mean amount of coffee dispensed by the machine is less than 12 fluid ounces.

Problem 2:

The director of a fitness center claims that the average time members spend at the gym per visit is 75 minutes. To test this claim, a trainer randomly selects 10 members and records the length of their next visit in minutes:

$80,65,90,70,75,85,70,60,95,72$

Do these data provide convincing evidence at the α = 0.10 level that the true mean time spent at the gym is different from 75 minutes?

Solution:

STATE:

• Parameter: Let μ = the true mean time (in minutes) that members spend at the gym per visit.

• Hypotheses:

  • H₀: μ = 75

  • Hₐ: μ \neq 75

• Significance Level: α = 0.10

PLAN:

• Procedure: We will perform a one-sample t-test for a population mean.

• Conditions:

  1. Random: The problem states the trainer "randomly selects 10 members."

  2. 10% Condition: 10 members is likely less than 10% of all members of the fitness center.

  3. Normal/Large Sample: The sample size (n=10) is small. We must check a graph of the data. A boxplot of the data shows no strong skewness or outliers, so we can proceed. (First, we enter the data into L1 and find summary statistics: x̄ = 76.2, sₓ = 11.13 minutes).

DO:

• Specific Formula:$t=(\bar{x}-{\mu }_0)/(s_x/\surd n)$

• Calculations:

  $t=(76.2-75)/(11.13/\surd 10)=1.2/3.5196=0.341$

• P-value:

  • Degrees of freedom: $df=n-1=10-1=9$.

  • This is a two-tailed test, so the p-value is $2\ast P(t\ge 0.341)$ with df=9. Using a calculator ($2\ast tcdf(0.341,1E99,9)$), we find the p-value is approximately 0.741.

CONCLUDE:

• Decision: Because our p-value of 0.741 is greater than our significance level of α = 0.10, we fail to reject the null hypothesis.

• Context: We do not have convincing evidence that the true mean time members spend at the gym per visit is different from 75 minutes.

Common Mistakes to Avoid

• Using z instead of t: This is the most common error. If you do not know the population standard deviation (σ) and are using the sample standard deviation (sₓ) instead, you must use a t-test, regardless of the sample size. The z-test is only for means when σ is known (which is very rare) or for proportions.

• Incorrect Degrees of Freedom: Always use $df=n-1$ for a one-sample t-test. Using $n$ will give you an incorrect p-value.

• Misinterpreting the P-value: Do not say "The p-value is the probability that the null hypothesis is true." The correct interpretation is: "The p-value is the probability of getting our sample result (or one more extreme) if the null hypothesis were true." It's a conditional probability about the data, not a probability about the hypothesis.

• "Accepting" the Null Hypothesis: You must never state that you "accept H₀." A large p-value simply means you don't have enough evidence to reject H₀. It does not prove that H₀ is true. The correct phrasing is always "fail to reject H₀."

• Forgetting Context in the Conclusion: Your final conclusion must answer the original question. Don't just say "We reject the null hypothesis." Explain what that means in terms of coffee machines, gym times, or whatever the problem is about.