Hypothesis Testing Two Populations Dependent Using Statcrunch Example 1

Main ideas / lessons

• The video explains hypothesis testing for two dependent populations using a paired (matched) samples t-test.
• “Dependent” arises because the data are matched by the year the awards were given (each actress age is paired with an actor age from the same award year).
• The goal is to test a claim about the mean difference between paired values.

Example setup (paired data)

• Sample data:
  • Ages of actresses who won Best Actress (for a set of years)
  • Ages of actors who won Best Actor (same years)
• Pairing rule: ages are matched by award year.
• Define the difference for each matched pair:
  • ( D = ) (actress age) (-) (actor age)
• Let:
  • ( \mu_D ) = mean of all such differences in the population

Hypothesis test (Part A: left-tailed test at ( \alpha = 0.05 ))

Symbolic hypotheses

• Original claim: actresses are generally younger than actors (\Rightarrow) the mean difference is less than zero

  • Alternative hypothesis:
    • ( H_a: \mu_D < 0 )
  • If the original claim were false:
  • ( \mu_D \ge 0 )
  • Null hypothesis (always includes equality):
  • ( H_0: \mu_D = 0 )
  • Test type:
  • Left-tail (one-tailed) because ( \mu_D ) is tested to be less than 0.

Using StatCrunch (method)

Because the populations are dependent/matched, use:

• Stat → t Stats → Paired

Enter data columns:

• Column 1: actresses ages
• Column 2: actors ages

Configure:

• Hypothesis test
• Confidence level / critical level options as needed
• Significance level: 0.05

Reported output used:

• Test statistic: approximately -2.38 (rounded to 2 decimals)
• p-value: approximately 0.021 (rounded to 3 decimals)

Decision rule and conclusion

Compare p-value (0.021) to significance level (0.05):

• Since (0.021 < 0.05), reject (H_0).

Conclusion (in words): There is sufficient evidence that actresses are generally younger than actors when they won the awards.

Confidence interval (Part B)

Confidence level selection

Because this was a one-tailed (left-tail) test with (\alpha = 0.05), the confidence level is:

• ( 1 - 2\alpha = 1 - 2(0.05) = 0.90 )

So the interval is a 90% confidence interval.

Using StatCrunch (method)

• Return to StatCrunch’s paired t settings
• Set confidence level to 90% (0.90)
• Compute confidence interval

Confidence interval results

Reported interval (rounded):

• Lower limit: -18.0
• Upper limit: -2.4

Key interpretation:

• The interval contains only negative values.
• Since negative-only values imply ( \mu_D < 0 ), this matches the Part A conclusion:
  • Reject (H_0) (which states ( \mu_D = 0 )).

Speaker(s) / sources

• No named speakers are identified in the subtitles.
• Source referenced: StatCrunch software (used to perform the paired t-test and confidence interval).