Listed below are the lead concentrations​ (in mu g divided by g​) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States. 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 mu g divided by g.
2.97
6.50
5.97
5.50
20.49
7.51
12.03
20.49
11.54
17.53
 
Question content area bottom
Part 1
Identify the null and alternative hypotheses.
Upper H 0​: 
mu
equals
  
14.0
Upper H 1​: 
mu
less than
  
14.0
​(Type integers or decimals. Do not​ round.)
Part 2
Identify the test statistic.
  
negative 1.44
​(Round to two decimal places as​ needed.)
Part 3
Identify the​ P-value.
  
enter your response here
​(Round to three decimal places as​ needed.)

### 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 \( \mu g/g \).

---

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.

Listed below are the lead concentrations​ (in mu g divided by g​) measured in different Ayurveda medicines. Ayurveda is a traditional medical system commonly used in India. The lead concentrations listed here are from medicines manufactured in the United States. 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 mu g divided by g.
2.97, 6.50, 5.97, 5.50, 20.49, 7.51, 12.03, 20.49, 11.54, 17.53

Learn how to test the claim that the mean lead concentration in Ayurveda medicines is less than 14.0 μg/g using a one-sample t-test. This problem involves hypothesis testing, statistical formulas, and key concepts in inferential statistics. Suitable for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation