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After reading the scenario below, write out which statistical test you would use (be very specific; just stating a general type of test or a name that could be considered a group of tests will be marked as incorrect).

"Researchers have noticed that there appears to be a regular, year-by-year increase in the average IQ for the general population. To evaluate the size of the effect, a researcher obtained a 10-year-old IQ test that was standardized to produce a mean IQ of [tex]\mu = 100[/tex] for the population 10 years ago. The test was then given to a sample of [tex]n = 64[/tex] of today’s adults. The average score for the sample was [tex]M = 107[/tex] with [tex]s = 12[/tex]. Based on the sample, is the average IQ for today’s population significantly different from the average 10 years ago when the test would have produced a mean of [tex]\mu = 100[/tex]?"

Answer :

To evaluate whether the average IQ for today's population is significantly different from the average 10 years ago, an appropriate statistical test to use is a one-sample t-test.

Here's how you can carry out the analysis step-by-step:

  1. Identify the Hypotheses:

    • Null Hypothesis ([tex]H_0[/tex]): The mean IQ of today's population is 100, the same as 10 years ago. Mathematically, [tex]H_0: \mu = 100[/tex].
    • Alternative Hypothesis ([tex]H_a[/tex]): The mean IQ of today's population is different from 100. Mathematically, [tex]H_a: \mu \neq 100[/tex].

  2. Determine the Test to Use:

    • Since we know the sample mean ([tex]M = 107[/tex]), the sample standard deviation ([tex]s = 12[/tex]), and the sample size ([tex]n = 64[/tex]), and we want to compare it to a known population mean ([tex]\mu = 100[/tex]), a one-sample t-test is appropriate.

  3. Calculate the Test Statistic:

    • The formula for the one-sample t-test is:
      [tex]t = \frac{M - \mu}{s / \sqrt{n}}[/tex]
    • Plugging in the values:
      [tex]t = \frac{107 - 100}{12 / \sqrt{64}} = \frac{7}{1.5} = 4.67[/tex]

  4. Determine the Degrees of Freedom:

    • For a one-sample t-test, the degrees of freedom (df) is [tex]n - 1[/tex]. Thus, [tex]df = 64 - 1 = 63[/tex].

  5. Find the Critical t-value and Make a Decision:

    • Using a t-distribution table or software, consult the t-value at [tex]df = 63[/tex] with your desired level of significance (usually [tex]b1 = 0.05[/tex] for a two-tailed test).
    • If the calculated t-statistic exceeds the critical t-value from the table, reject the null hypothesis.

  6. Conclusion:

    • Given the calculated t-value of 4.67, which is likely to exceed the critical t-value for [tex]df = 63[/tex], you would reject the null hypothesis.
    • Therefore, the sample provides sufficient evidence to conclude that the average IQ for today's population is significantly different from the average 10 years ago.

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