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### Problem Summary:
You are given a set of lead concentration measurements (in µg/g) from different Ayurveda medicines made in the U.S. and asked to test the claim that the **mean lead concentration** is less than **14.0 µg/g**. A **0.01 significance level** is specified for the test.

Here are the values provided:
\[
3.03, 6.50, 6.00, 5.50, 20.52, 7.54, 12.01, 20.50, 11.51, 17.50
\]

### Steps to Complete:

#### 1. **Identify the null and alternative hypotheses**:
   - **Null hypothesis (H₀)**: The mean lead concentration \( \mu \) is equal to 14.0 µg/g. This can be written as:
     \[
     H_0: \mu = 14.0
     \]
   - **Alternative hypothesis (H₁)**: The mean lead concentration \( \mu \) is less than 14.0 µg/g (since we are testing if it is less). This can be written as:
     \[
     H_1: \mu < 14.0
     \]

#### 2. **Test Statistic**:
   This involves computing the **sample mean** \( \bar{x} \), the **sample standard deviation** \( s \), and then calculating the **t-statistic** using the formula:
   \[
   t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
   \]
   where:
   - \( \bar{x} \) = sample mean
   - \( \mu \) = hypothesized population mean (14.0)
   - \( s \) = sample standard deviation
   - \( n \) = sample size

#### 3. **P-Value**:
   Once the t-statistic is calculated, the **p-value** can be found using a t-distribution table or software, based on the degrees of freedom \( n - 1 \).

#### 4. **Conclusion**:
   Based on the p-value, you will compare it to the **0.01 significance level**:
   - If \( p \leq 0.01 \), reject the null hypothesis.
   - If \( p > 0.01 \), fail to reject the null hypothesis.

I will now calculate the test statistic, p-value, and conclusion. Let me do that.Here are the results:

1. **Sample Mean**: \( \bar{x} = 11.06 \)
2. **Sample Standard Deviation**: \( s = 6.46 \)
3. **t-Statistic**: \( t = -1.44 \)
4. **P-value**: \( p = 0.092 \)

### Conclusion:
- Since the **p-value (0.092)** is greater than the **significance level (0.01)**, we **fail to reject the null hypothesis**. There is **not sufficient evidence** at the 0.01 significance level to support the claim that the mean lead concentration for Ayurveda medicines is less than 14.0 µg/g.

Would you like further details or clarifications? Here are some related questions you might explore:

1. How do you interpret a p-value in hypothesis testing?
2. What does the t-statistic represent in a t-test?
3. Why do we use a significance level of 0.01, and what does it mean?
4. How would increasing the sample size affect the results?
5. What are the assumptions underlying the t-test in this scenario?

**Tip**: In hypothesis testing, always make sure the sample data satisfies the test's assumptions, such as normality for the t-test.

Listed below are the lead concentrations (in µg/g) measured in different Ayurveda medicines. Assume that a simple random sample has been selected. Use a 0.01 significance level to test the claim that the mean lead concentration for all such medicines is less than 14.0 µg/g.

Perform a hypothesis test at the 0.01 significance level to determine if the mean lead concentration in Ayurveda medicines is less than 14.0 µg/g. Learn about hypothesis testing, t-statistics, and p-values in this statistical analysis.

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