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Question 9, 3.1.20

HW Score: [tex]$82.5 \%, 8.25$[/tex] of 10 points

Part 2 of 3

Points: 0.25 of 1

A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is [tex]$65 \%$[/tex]. During one semester, 200 students enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 147 passed the course. The instructor carried out a hypothesis test and found that the observed value of the test statistic was significant. The [tex]$p$[/tex]-value associated with this test statistic is 0.0059.

Explain the meaning of the [tex]$p$[/tex]-value in this context. Based on this result, should the instructor believe the success rate has improved?

State the hypotheses that were used for the test. Choose the correct answer below:

A. [tex]$H_0: p \ \textless \ 0.65$[/tex] and [tex]$H_a: p \ \textgreater \ 0.65$[/tex]

B. [tex]$H_0: p = 0.65$[/tex] and [tex]$H_a: p \ \textgreater \ 0.65$[/tex]

C. [tex]$H_0: p = 0.65$[/tex] and [tex]$H_a: p \ \textless \ 0.65$[/tex]

D. [tex]$H_0: p \ \textgreater \ 0.65$[/tex] and [tex]$H_a: p \ \textless \ 0.65$[/tex]

E. [tex]$H_0: p = 0.65$[/tex] and [tex]$H_a: p \neq 0.65$[/tex]

Explain the meaning of the [tex]$p$[/tex]-value in this context. Select the correct choice below and fill in the answer box within your choice:

(Type an integer or a decimal. Do not round.)

A. The probability of 147 or more introductory chemistry students passing out of a random sample of 200 students is 0.65, assuming the population proportion is greater than 0.65.

B. The probability of 147 or fewer introductory chemistry students passing out of a random sample of 200 students is [tex]$\square$[/tex], assuming the population proportion is 0.65.

C. The probability of 147 or more introductory chemistry students passing out of a random sample of 200 students is [tex]$\square$[/tex], assuming the population proportion is 0.65.

D. The probability of 147 or fewer introductory chemistry students passing out of a random sample of 200 students is [tex]$\square$[/tex], assuming the population proportion is less than 0.65.

Sagot :

GinnyAnsweridn
### Step-by-Step Solution:

1. State the hypotheses for the test:
- The null hypothesis [tex]\(H_0\)[/tex] is the assumption that the passing rate has not changed.
- The alternative hypothesis [tex]\(H_a\)[/tex] is the assumption that the passing rate has improved.

Given the problem's context, we set up the hypotheses as follows:
[tex]\[ H_0: p = 0.65 \quad \text{and} \quad H_a: p > 0.65 \][/tex]
Thus, the correct answer is:
- B. [tex]\( H_0: p=0.65 \)[/tex] and [tex]\( H_a: p>0.65 \)[/tex]

2. Explain the meaning of the [tex]\( p \)[/tex]-value in this context:
- The [tex]\( p \)[/tex]-value tells us the probability of observing a sample proportion as extreme as, or more extreme than, the one observed (147 out of 200 students passing), assuming that the null hypothesis [tex]\(H_0\)[/tex] is true.

Specifically, it is the probability of 147 or more students passing given that the population proportion [tex]\( p \)[/tex] is actually 0.65.

Given the [tex]\( p \)[/tex]-value of 0.0059, the correct interpretation in the context of the problem is:
- C. The probability of 147 or more introductory chemistry students passing out of a random sample of 200 students is 0.0059, assuming the population proportion is 0.65.

3. Determine if the instructor should believe the success rate has improved:
- To make this decision, we compare the [tex]\( p \)[/tex]-value to the significance level α. Common practice is to use α = 0.05.
- If the [tex]\( p \)[/tex]-value is less than α, we reject the null hypothesis [tex]\(H_0\)[/tex] in favor of the alternative hypothesis [tex]\(H_a\)[/tex].

In this case:
- The [tex]\( p \)[/tex]-value is 0.0059, which is less than 0.05.
- Therefore, we reject [tex]\( H_0 \)[/tex] and accept [tex]\( H_a \)[/tex], meaning we have enough evidence to support the claim that the success rate has improved.

The answer is:
- Based on the [tex]\( p \)[/tex]-value of 0.0059, the instructor should believe that the success rate has improved.

### Summary:
1. Hypotheses: B. [tex]\( H_0: p=0.65 \)[/tex] and [tex]\( H_a: p>0.65 \)[/tex]
2. [tex]\( p \)[/tex]-value interpretation: C. The probability of 147 or more introductory chemistry students passing out of a random sample of 200 students is 0.0059, assuming the population proportion is 0.65.
3. Conclusion: The instructor should believe that the success rate has improved.
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