To test the claim that the mean lead concentration for all such traditional medicines is less than [tex]\( 13 \, \mu \text{g} / \text{g} \)[/tex], we need to perform a hypothesis test. Here is the step-by-step solution:
### Step 1: Formulate Hypotheses
We define the null hypothesis ([tex]\( H_0 \)[/tex]) and the alternative hypothesis ([tex]\( H_1 \)[/tex]) as follows:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean lead concentration is [tex]\( 13 \, \mu \text{g} / \text{g} \)[/tex].
[tex]\[ H_0: \mu = 13 \, \mu \text{g} / \text{g} \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean lead concentration is less than [tex]\( 13 \, \mu \text{g} / \text{g} \)[/tex].
[tex]\[ H_1: \mu < 13 \, \mu \text{g} / \text{g} \][/tex]
We will use a significance level ([tex]\( \alpha \)[/tex]) of 0.10.
### Step 2: Calculate the Test Statistic
The test statistic for a one-sample t-test is calculated using the formula:
[tex]\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \][/tex]
Where:
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu_0 \)[/tex] is the claimed population mean.
- [tex]\( s \)[/tex] is the sample standard deviation.
- [tex]\( n \)[/tex] is the sample size.
Based on the numerical results:
- Sample mean ([tex]\( \bar{x} \)[/tex]): 3.789
- Sample standard deviation ([tex]\( s \)[/tex]): 6.752
- Sample size ([tex]\( n \)[/tex]): 7
- Claimed mean ([tex]\( \mu_0 \)[/tex]): 13
The calculated test statistic is:
[tex]\[ t = -3.609 \][/tex]
### Step 3: Determine the Critical Value
For a one-tailed test with a significance level of 0.10 and degrees of freedom [tex]\( df = n - 1 = 6 \)[/tex], we look up the critical t-value in the t-distribution table or compute it using statistical software. The critical t-value for [tex]\( \alpha = 0.10 \)[/tex] with 6 degrees of freedom is:
[tex]\[ t_{\text{critical}} = -1.440 \][/tex]
### Step 4: Calculate the P-value
The p-value associated with the calculated t-score can be found using the cumulative distribution function (CDF) of the t-distribution:
[tex]\[ \text{p-value} = 0.0056 \][/tex]
### Step 5: Make a Decision
We compare the p-value with the significance level ([tex]\( \alpha \)[/tex]):
- If the p-value is less than [tex]\( \alpha \)[/tex], we reject the null hypothesis.
- If the p-value is greater than or equal to [tex]\( \alpha \)[/tex], we do not reject the null hypothesis.
In this case:
[tex]\[ \text{p-value} = 0.0056 < 0.10 \][/tex]
Since the p-value is less than the significance level, we reject the null hypothesis.
### Conclusion
At the 0.10 significance level, we have sufficient evidence to reject the null hypothesis and support the claim that the mean lead concentration in traditional medicines is less than [tex]\( 13 \, \mu \text{g} / \text{g} \)[/tex].
