Question

A machine that puts corn flakes into boxes is adjusted to put an average of 15.9 ounces into each box, with standard deviation of 0.24 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.


State the null and alternate hypotheses.

H₀: σ² = 0.0576; H₁: σ² > 0.0576H₀: σ² < 0.0576; H₁: σ² = 0.0576    H₀: σ² = 0.0576; H₁: σ² < 0.0576H₀: σ² = 0.0576; H₁: σ² ≠ 0.0576

(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 > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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

Population variance = σ02 = 0.24^2 = 0.0576

Sample variance = s2 = 0.37^2 = 0.1369

Answer (i)

The value of the level of significance is 0..01

The null and alternate hypotheses are:

H0: σ2 = 0.0576; H1: σ2 > 0.0576

Answer (ii)

The sample test statistic χ2 = 26.14

Answer (iii)

Degrees of Freedom = n-1 = 12-1 = 11

P-value corresponding to χ2 = 26.14 and df = 11 is obtained using p-value calculator. Screenshot below:

The P-value of the sample test statistic = 0.0062

So, correct answer is 0.005 < P-value < 0.010

Answer (iv)

Since P-value = 0.0062 < α = 0.01, we reject null hypothesis

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

Answer (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.