Hypothesis testing for Population Means (Sigma known)

Lesson Goals

• Understand how to determine the test statistic when testing a population mean with known sigma.

• Utilize a rejection region to draw conclusions about hypothesis tests.

• Apply p-values to make informed decisions in hypothesis testing.

General Hypothesis Testing Steps

• Steps to follow in hypothesis testing:

  • Step 1: State the null (H0) and alternative (H1) hypotheses.

  • Step 2: Determine which distribution to use for the test statistic and state the significance level (alpha).

  • Step 3: Gather data and calculate necessary sample statistics.

  • Step 4: Draw a conclusion and interpret the decision.

Hypothesis Test for Population Mean with Sigma Known

• Investigate performing hypothesis tests for a population mean when sigma is known.

• Criteria for selecting the appropriate probability distribution:

  • Determine if the population standard deviation (sigma) is known.

  • If sigma is known, use a z test statistic or advanced methods, recognizing this is a rare scenario in practice.

Conditions for Using the Standard Normal Distribution

• Conditions that must be satisfied to use the standard normal distribution:

  1. All samples of a given size must have equal probability of being selected (simple random sampling).

  2. The population standard deviation (sigma) is known.

  3. Sample size (n) is at least 30 or the population distribution is approximately normal.

Test Statistic Formula

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  Z-test statistic formula for hypothesis testing with known sigma:[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} ]

  • Where:

    • ( \bar{X} ) = sample mean

    • ( \mu ) = hypothesized population mean from H0

    • ( \sigma ) = population standard deviation

    • ( n ) = sample size

Rejection Regions

• Standard process to determine if test statistic is statistically significant:

  • Based on:

    1. Type of hypothesis test

    2. Significance level (alpha)

  • Types of hypothesis tests:

    • Left-tailed: Alternative hypothesis uses "<" symbol.

    • Right-tailed: Alternative hypothesis uses ">" symbol.

    • Two-tailed: Alternative hypothesis uses "≠" symbol.

Constructing Rejection Regions

• Left-tailed test:

  • Rejection region in the left tail, equal to significance level alpha.

• Right-tailed test:

  • Rejection region in the right tail, equal to significance level alpha.

• Two-tailed test:

  • Rejection region divided evenly in two tails, each tail area = alpha/2.

Decision Rule

• Formulate decision rules based on the rejection region:

  • Reject H0 if test statistic ( Z ) falls within the rejection region:

    • Left-tailed: ( Z \leq -Z_{\alpha} )

    • Right-tailed: ( Z \geq Z_{\alpha} )

    • Two-tailed: ( |Z| \geq Z_{\alpha/2} )

P-Values

• More common method in research for drawing conclusions:

  • Definition: The probability of observing a sample statistic as extreme or more extreme than the sample data under the assumption the null hypothesis is true.

  • For left-tailed tests, find the probability of ( Z ) being less than or equal to the calculated value.

Conclusions Using P-Values

• Compare the calculated p-value to alpha:

  • If ( p \leq \alpha ): reject H0.

  • If ( p > \alpha ): fail to reject H0.

Summary of Steps for Performing a Hypothesis Test with Sigma Known

1. State null and alternative hypotheses.

2. Determine distribution for test statistic and level of significance (alpha); use standard normal distribution for known sigma.

3. Calculate test statistic: ( Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} ).

4. Draw conclusions using either rejection regions or p-value comparisons with alpha.