Question

The fan blades on commercial jet engines must be replaced when wear on these parts indicates too much variability to pass inspection. If a single fan blade broke during operation, it could severely endanger a flight. A large engine contains thousands of fan blades, and safety regulations require that variability measurements on the population of all blades not exceed σ² = 0.18 mm². An engine inspector took a random sample of 81 fan blades from an engine. She measured each blade and found a sample variance of 0.28 mm². Using a 0.01 level of significance, is the inspector justified in claiming that all the engine fan blades must be replaced?

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: σ² = 0.18; H₁: σ² ≠ 0.18H₀: σ² > 0.18; H₁: σ² = 0.18    H₀: σ² = 0.18; H₁: σ² < 0.18H₀: σ² = 0.18; H₁: σ² > 0.18

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a normal population distribution.We assume a exponential population distribution.    We assume a binomial population distribution.We assume a uniform population distribution.

(c) 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

(d) Based on your answers in parts (a) to (c), will you reject or fail to 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.    Since the P-value ≤ α, we reject the null hypothesis.Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 1% level of significance, there is insufficient evidence to conclude that the variance of measurements on the fan blades is higher than the specified amount. The inspector is not justified in claiming the blades must be replaced.At the 1% level of significance, there is sufficient evidence to conclude that the variance of measurements on the fan blades is higher than the specified amount. The inspector is justified in claiming the blades must be replaced.    At the 1% level of significance, there is sufficient evidence to conclude that the variance of measurements on the fan blades is higher than the specified amount. The inspector is not justified in claiming the blades must be replaced.At the 1% level of significance, there is insufficient evidence to conclude that the variance of measurements on the fan blades is higher than the specified amount. The inspector is justified in claiming the blades must be replaced.

(f) Find a 90% confidence interval for the population standard deviation. (Round your answers to two decimal places.)

lower limit	mm
upper limit	mm

Interpret the results in the context of the application.

We are 90% confident that σ lies above this interval.We are 90% confident that σ lies outside this interval.    We are 90% confident that σ lies below this interval.We are 90% confident that σ lies within this interval.

Answer 1

(a) The level of significance is 0.01 0.18 H00.18 Ho o (b) Excel add-in (Mega Stat) procedure: Step 1: In EXCEL, Select Add-Ins Mega Stat Hypothesis Tests. Step 2: Choose Chi-square Variance Test. Step 3: Choose Summary Input Step 4: Enter Bl:B3 under Input Range (Select ranges with label variance, n) Step 5: Enter 0.18 Under Hypothesized Variance Step 6: Choose greater than in alternative Step 7: Click OK > > Excel add-in (Mega Stat) output:

Chi-square Variance Test 0.1800 hypothesized variance 0.2800 observed variance of Sample 81 n 80 df 124.44 chi-square 0011 p-value (one-tailed, upper) The chi-square statistic for the sample is 124.44 80 Degrees of freedom We assume a nomal population distribution (c) P-value< 0.005 (d) Since the P-value sa, we reject the null hypothesis

(e) At the 1% level of significance, there is sufficient evidence to conclude that the variance of measurements on the fan blades is higher than the specified amount. The inspector is justified in claiming the blades must be replaced (f) Excel add-in (PH Stat) procedure: Step 1: In EXCEL, Select Add-Ins PH Stat> Confidence Intervals Estimate for the Population Variance Step 2: In Data, enter 81 as the Sample size, 0.53 as the Sample Standard deviation, and 90% as the Confidence Level Step 3: Click OK Excel add-in (PH Stat) output

Confidence Interval Estimate for the Population Variance Data Sample Size Sample Standard Deviation 81 0.53 Confidence Level 90% Intermediate Calculations Degrees of Freedom Sum of Squares Single Tail Area 80 22,472 0.05 Lower Chi-Square Value 60.3915 Upper Chi-Square Value 101.8795 Results Interval Lower Limit for Variance 0.2206 Interval Upper Limit for Variance 0.3721 Interval Lower Limit for Standard Deviation 0.4697 Interval Upper Limit for Standard Deviation 0.6100 Lower limit 0.47mm Upper limit 0.6 1mm We are 90% confident that o lies within this interval