Part 1: Hypotheses

We are testing the claim that the mean lead concentration for all such medicines is less than $14.0 \, \mu g/g$.

• Null Hypothesis $H_0$: $H_0: \mu = 14.0 \, \mu g/g$
• Alternative Hypothesis $H_1$: $H_1: \mu < 14.0 \, \mu g/g$

This is a one-tailed test, as we are interested in determining if the mean is less than 14.0.

Part 2: Test Statistic

You are given that the test statistic is $t = -1.44$. This value was likely calculated using the formula for the t-test statistic for a one-sample t-test:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where:

• $\bar{x}$ is the sample mean,
• $\mu_0$ is the hypothesized population mean (14.0 in this case),
• $s$ is the sample standard deviation,
• $n$ is the sample size.

Part 3: P-value

The P-value is the probability that the test statistic $t = -1.44$ would occur under the null hypothesis. Since this is a left-tailed test, we need to find the probability that a t-statistic is less than $-1.44$ for a certain number of degrees of freedom.

We can calculate the P-value using a t-distribution table or a statistical software. Let's break it down:

• The sample size $n = 10$,
• Degrees of freedom $df = n - 1 = 9$.

Using a t-distribution table or calculator, the P-value corresponding to $t = -1.44$ with 9 degrees of freedom is approximately:

$P\text{-value} \approx 0.089$

Conclusion:

• If the P-value $\approx 0.089$ is greater than the significance level $\alpha = 0.01$, we fail to reject the null hypothesis.
• This means there is not enough evidence at the $0.01$ significance level to conclude that the mean lead concentration is less than 14.0 $\mu g/g$.

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Would you like more details or clarification on any part?

5 Related Questions:

1. How is the t-test statistic calculated for one-sample tests?
2. What assumptions must be met for conducting a t-test?
3. How can we interpret the P-value in hypothesis testing?
4. How would the conclusion change if the P-value were smaller than 0.01?
5. What is the difference between a one-tailed and two-tailed t-test?

Tip:

Always ensure your data meets the assumptions of normality and random sampling when performing a t-test to ensure valid results.