Question

question 10
not yet
answered
points out of
1.00
flag
question
a manufacturer must test that his bolts are 3 cm long when they come off the assembly line. after sampling 123
he must measure his machines if the bolts are too long or too short. he calculates the sample mean to be 2.87 cm
randomly selected bolts off the assembly line. he knows that the population standard deviation is 0.58 cm. assuming a level of significance
is 0.05 is used for the hypothesis test.
which of the following represents the alternative hypothesis?
□ a μ = 3
□ b. μ ≤ 3
□ c μ ≠ 3
question 11
not yet
answered
points out of
1.00
flag
question
is this a one tailed or a two tailed test?
□ a. one tailed test
□ b. two tailed test
question 12
not yet
answered
points out of
1.00
flag
question
what is the value of the test statistic? round your answer to two decimal places.
answer: -2.51
question 13
not yet
answered
points out of
1.00
flag
question
what is the critical value?
answer with a positive number only.
answer: 1.96
question 14
not yet
answered
points out of
1.00
flag
question
what is the p - value?
answer: 0.012
question 15
not yet
answered
points out of
1.00
flag
question
based off of either a rejection region or the p - value, make a conclusion.
□ a. reject the null hypothesis
□ b. fail to reject the null hypothesis

Explanation:

Step1: Hypothesis Testing Basics

The null hypothesis \(H_0:\mu = 3\) (the claim that bolts are 3 cm long). The alternative hypothesis \(H_1:\mu
eq3\) (since we are testing if bolts are not 3 cm long, either too long or too short).

Step2: One - Tailed vs Two - Tailed Test

A two - tailed test is used when the alternative hypothesis is \(\mu
eq\) a value. Here, since we are concerned with bolts being either too long or too short (deviation in both directions from 3 cm), it is a two - tailed test.

Step3: Test Statistic Calculation

The formula for the z - test statistic is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Given \(\bar{x} = 2.87\), \(\mu = 3\), \(\sigma=0.58\), \(n = 123\).

$$z=\frac{2.87 - 3}{\frac{0.58}{\sqrt{123}}}=\frac{- 0.13}{\frac{0.58}{11.09}}=\frac{-0.13}{0.0523}\approx - 2.51$$

Step4: Critical Value

For a two - tailed test with \(\alpha = 0.05\), the critical values are \(z_{\alpha/2}\). Using standard normal distribution tables, \(z_{0.025}=1.96\) (positive value as asked).

Step5: p - Value

For \(z=-2.51\), the p - value for a two - tailed test is \(2P(Z < - 2.51)\). Using standard normal distribution tables, \(P(Z < - 2.51)=0.006\), so \(p - value=2\times0.006 = 0.012\)

Step6: Conclusion

Since \(|z|=2.51>1.96\) (rejection region approach) or \(p - value = 0.012<0.05\) (\(\alpha\)), we reject the null hypothesis.

Answer:

Question 10: C. \(\mu
eq3\)
Question 11: B. Two Tailed Test
Question 12: - 2.51
Question 13: 1.96
Question 14: 0.012
Question 15: A. Reject the Null Hypothesis