According to Forbes, the average age of millionaires in the US 10 years ago was 62 years old. Since then, the number of millionaires in the US has grown to over 17 million. One prevalent belief among economists is that the average age of millionaires today has decreased from the average age of millionaires 10 years ago. To investigate this belief, a random sample of 50 millionaires from the US was selected and their ages were recorded. Here are the summary statistics:

| Statistic | [tex]$n$[/tex] | [tex]$\bar{x}$[/tex] | [tex]$s_x$[/tex] | Min | [tex]$Q_1$[/tex] | Med | [tex]$Q_3$[/tex] | Max |
|-----------|----|----------|------|-----|------|-----|------|-----|
| Value | 50 | 51 | 10.2 | 18 | 38 | 50.5 | 72 | 99 |

Based on this sample, is there convincing evidence that the true mean age of millionaires in the US is less than 62 years old? Use [tex]$\alpha=0.01$[/tex]. Provide statistical evidence to support your answer.



Answer :

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.