Find the critical​ value(s) for the type of​ t-test with level of significance

alphaα

and sample size n.

Upper H 0H0​:

muμless than or equals≤​20,

alphaαequals=0.100.10​,

nequals=2020

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Part 1

A.negative 2.093

negative 2.093−2.093

B.negative 2.861

negative 2.861−2.861

C.negative 1.729

negative 1.729−1.729

D.negative 1.328

negative 1.328

To find the critical value for the given one-sample t-test, we will use the following information:

- **Null Hypothesis (\( H_0 \))**: \( \mu \leq 20 \)
- **Significance level (\( \alpha \))**: 0.10
- **Sample size (\( n \))**: 20
- **Degrees of freedom (\( df \))**: \( n - 1 = 19 \)

Since the null hypothesis is \( \mu \leq 20 \), we are dealing with a one-tailed test (specifically a left-tailed test). We need to find the t-value that corresponds to the given level of significance (\( \alpha = 0.10 \)) for 19 degrees of freedom.

### Steps to Find the Critical Value:
1. **Determine the degrees of freedom**: \( df = 20 - 1 = 19 \).
2. **Locate the critical t-value for a one-tailed test with \( \alpha = 0.10 \) and \( df = 19 \)**. 

For a one-tailed t-test with \( \alpha = 0.10 \) and 19 degrees of freedom, we can use a t-distribution table or statistical software to find the critical value.

### Finding the t-value:

Looking up the value in a t-distribution table, the critical value (negative since it's a left-tailed test) for \( \alpha = 0.10 \) and \( df = 19 \) is approximately:

\[
t_{0.10, 19} \approx -1.328
\]

Thus, the correct critical value is:

**D. \( -1.328 \)**

Would you like more details or have any questions?

Here are some related questions to explore:
1. How do you determine whether to use a one-tailed or two-tailed t-test?
2. What is the difference between a one-sample t-test and a two-sample t-test?
3. How does the sample size affect the critical value in a t-test?
4. What are degrees of freedom, and why are they important in statistical tests?
5. How does changing the significance level \( \alpha \) affect the critical value?
6. What assumptions must be met to perform a t-test?
7. How do you interpret the result of a t-test in context?
8. What is the difference between a t-test and a z-test?

**Tip:** When using a t-distribution table, always ensure you are using the correct tail and degrees of freedom for your specific test.

Learn how to find the critical value for a one-sample t-test with a significance level α = 0.10 and sample size n = 20. Understand the hypothesis testing process, degrees of freedom calculation, and application of the t-distribution. Suitable for advanced undergraduate students or anyone studying hypothesis testing in statistics.

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