This is a hypothesis testing problem involving two independent samples. We are asked to determine if there is a significant difference in the mean blood lead levels between schoolchildren and street child vendors. Let me outline the steps and solve the problem.

Given Data:

• Group 1: Schoolchildren

  • $n_1 = 414$
  • $\bar{X}_1 = 14.1 \, \mu g/dl$ (mean blood lead level)
  • $s_1 = 5.3 \, \mu g/dl$ (standard deviation)

• Group 2: Street child vendors

  • $n_2 = 101$
  • $\bar{X}_2 = 15.0 \, \mu g/dl$ (mean blood lead level)
  • $s_2 = 5.5 \, \mu g/dl$ (standard deviation)

• Significance level: $\alpha = 0.01$

We will use a two-sample t-test assuming equal population variances.

Hypotheses:

• Null Hypothesis $H_0: \mu_1 = \mu_2$ (There is no difference in mean blood lead levels between the two groups.)
• Alternative Hypothesis $H_1: \mu_1 \neq \mu_2$ (There is a difference in mean blood lead levels.)

Step 1: Calculate the test statistic

For a two-sample t-test with equal variances, the formula for the test statistic is:

$t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Where:

• $s_p$ is the pooled standard deviation, calculated as:

$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

Step 2: Degrees of freedom

The degrees of freedom (df) for this test is:

$df = n_1 + n_2 - 2$

Now, let's compute these values.The computed values are as follows:

• The pooled standard deviation $s_p = 5.34 \, \mu g/dl$
• The test statistic $t = -1.52$
• The degrees of freedom $df = 513$

Step 3: Determine the critical value or p-value

Using a two-tailed test with $\alpha = 0.01$ and degrees of freedom $df = 513$, we can find the critical t-value from the t-distribution table or calculate the p-value. I'll now compute the p-value for this test.The p-value for the test is approximately $0.129$.

Step 4: Conclusion

Since the p-value $0.129$ is greater than the significance level $\alpha = 0.01$, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude a significant difference in the mean blood lead levels between schoolchildren and street child vendors at the 1% significance level.

Would you like further details on any step or have any questions?

Related Questions:

1. What is the impact of changing the significance level $\alpha$ on the results?
2. How would the conclusion change if population variances were unequal?
3. What assumptions are made in a two-sample t-test with equal variances?
4. How is the pooled standard deviation used in hypothesis testing?
5. Can you apply a one-tailed test in this scenario?

Tip:

When conducting hypothesis tests, always check the assumptions underlying the test (such as normality and equal variances) to ensure valid results.