Question

A machine that puts corn flakes into boxes is adjusted to put an average of 15.5 ounces into each box, with standard deviation of 0.21 ounce. If a random sample of 12 boxes gave a sample standard deviation of 0.37 ounce, do these data support the claim that the variance has increased and the machine needs to be brought back into adjustment? (Use a 0.01 level of significance.)

(i) Give the value of the level of significance.

0.01

State the null and alternate hypotheses.

*H*_{0}: *σ*^{2} = 0.0441;
*H*_{1}: *σ*^{2} ≠ 0.0441

*H*_{0}: *σ*^{2} < 0.0441;
*H*_{1}: *σ*^{2} =
0.0441

*H*_{0}: *σ*^{2} = 0.0441;
*H*_{1}: *σ*^{2} > 0.0441

*H*_{0}: *σ*^{2} = 0.0441;
*H*_{1}: *σ*^{2} < 0.0441

(ii) Find the sample test statistic. (Round your answer to two
decimal places.)

_____

(iii) Find or estimate the *P*-value of the sample test
statistic.

*P*-value > 0.100

0.050 < *P*-value <
0.100

0.025 < *P*-value < 0.050

0.010 < *P*-value < 0.025

0.005 < *P*-value < 0.010

*P*-value < 0.005

(iv) Conclude the test.

Since the *P*-value ≥ *α*, we fail to reject the
null hypothesis.

Since the *P*-value < *α*, we reject the null
hypothesis.

Since the *P*-value < *α*, we fail to reject
the null hypothesis.

Since the *P*-value ≥ *α*, we reject the null
hypothesis.

(v) Interpret the conclusion in the context of the application.

At the 1% level of significance, there is sufficient evidence to conclude that the variance has increased and the machine needs to be adjusted.

At the 1% level of significance, there is insufficient evidence to conclude that the variance has increased and the machine needs to be adjusted.

Answer #1

The statistical software output for this problem is :

(i)

*H*_{0}: *σ*^{2} = 0.0441;
*H*_{1}: *σ*^{2} > 0.0441

(ii)

sample test statistic = 34.15

(iii)

*P*-value < 0.005

(iv)

Since the *P*-value < *α*, we reject the null
hypothesis.

(v)

At the 1% level of significance, there is sufficient evidence to conclude that the variance has increased and the machine needs to be adjusted.

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