Let's analyze this question step-by-step:
1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The true mean age of millionaires today is 62 years (i.e., [tex]\(\mu = 62\)[/tex]).
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The true mean age of millionaires today is less than 62 years (i.e., [tex]\(\mu < 62\)[/tex]).
2. Given Data:
- Sample size ([tex]\(n\)[/tex]): 50
- Sample mean ([tex]\(\bar{x}\)[/tex]): 51 years
- Sample standard deviation ([tex]\(s_x\)[/tex]): 10.2 years
- Population mean 10 years ago ([tex]\(\mu_0\)[/tex]): 62 years
- Significance level ([tex]\(\alpha\)[/tex]): 0.01
3. Formulate the Test Statistic:
We will use a one-sample t-test since the population standard deviation is unknown and the sample size is relatively small.
The test statistic for the t-test is calculated using the formula:
[tex]\[
t = \frac{\bar{x} - \mu_0}{\frac{s_x}{\sqrt{n}}}
\][/tex]
Plugging in the values:
[tex]\[
t = \frac{51 - 62}{\frac{10.2}{\sqrt{50}}}
\][/tex]
From the given results, the test statistic is [tex]\(-7.626\)[/tex].
4. Determine the Critical Value:
The critical value for a left-tailed t-test at [tex]\(\alpha = 0.01\)[/tex] with [tex]\(df = n - 1 = 50 - 1 = 49\)[/tex] degrees of freedom is obtained from t-distribution tables.
From the given results, the critical value is [tex]\(-2.405\)[/tex].
5. Decision Rule:
Compare the test statistic with the critical value:
- If [tex]\( t \)[/tex] is less than the critical value, we reject the null hypothesis [tex]\(H_0\)[/tex].
- If [tex]\( t \)[/tex] is greater than or equal to the critical value, we fail to reject the null hypothesis [tex]\(H_0\)[/tex].
6. Conclusion:
In this case, the test statistic [tex]\( t = -7.626 \)[/tex] is less than the critical value [tex]\( -2.405 \)[/tex].
Therefore, we reject the null hypothesis [tex]\(H_0\)[/tex].
This provides strong evidence at the [tex]\(\alpha = 0.01\)[/tex] significance level to conclude that the true mean age of millionaires in the US is less than 62 years.
