Question

It is essential in the manufacture of machinery to utilize parts that conform to specifications. In the past, diameters of the ball-bearings produced by a certain manufacturer had a variance of 0.01 cm. To cut costs, the manufacturer instituted a less expensive production method. The variance of the diameter of 20 randomly samples bearings produced by the new process was 0.015 cm. Use α = 0.05 and assume that the diameters are normally distributed.

a) Use hypothesis testing to test if the data provide sufficient evidence to indicate that the diameters of ball‐bearings produced by the new process are more variable than those produced by the old process.

b) Use the confidence interval to solve (a).

c) Evaluate the P‐value of the analysis.

d) Calculate the probability to detect if diameter exceeds by 25% the one produced by new process using the same sample size.

e) Calculate the sample size required to detect a difference of 25% in (c) with a probability of at least 80%

Answer 1

a)

Ho :sigma = 0.01

Ha: simga > 0.01

s= sqrt(variance) =sqrt(0.015) = 0.12247

TS = (1-s)S^2/sigma^2

p-value = 0.074 > alpha

hence we fail reject the null hypothesis

we conclude that the data does not provide enough evidence to indicate that variance has increased.

b)

confidence interval for variance = (0.0087 ,0.0320)

since 0.01 is present in confidence interval, the data does not provide enough evidence to indicate that variance has increased.

c)

p-value = P(X^> TS) ==CHISQ.DIST.RT(28.5,19)

= 0.074

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