A local manufacturer of wooden furniture orders timber from nearby mills, to mak

Question

A local manufacturer of wooden furniture orders timber from nearby mills, to make their various products such as tables, chairs, and bedframes. One of the woodworkers there expresses their dissatisfaction with the quality of the timber supplied by the mills, and states that in their opinion, over three-quarters of timber pieces of a certain size range contain unworkable flaws. a) Test this claim, at the 95% level of confidence, against a subsequent random sample of 40 timber pieces in this size range, of which 35 pieces contain unworkable flaws. Use the critical-value method. b) Use the p-value method to determine if the result from Part (a) would be any different at alpha = 0.001, 0.01, or 0.10

Explanation / Answer

Given that, population proportion of a timber pieces containing unworkable flaws is P = 3/4 = 0.75.

In the given sample of size n = 40, x = 35 are having unworkable flaws.
Hence sample proportion p = x/n = 35/40 = 0.875.

(a)

We want to test hypothesis; H0: P = 0.75 vs Ha: P 0.75,

Let the test statistic for testing population proportion is given by,

z = p - P / sqrt ( P * (1 - P) / n ) = 1.8257

hence z-cal = 1.8257,

Now let confidence level = c = 95%, Hence z-critical value = 1.96.

Since z-cal (1.8257) is less than z-critical (1.96) hence we do not reject the null hypothesis.

Hence there is sufficient evidence that over three-quarters of timber pieces of a certain size range contain flaws.

(b)

Now p-value is given by, p-value = 2 * P(Z > z-cal ) = 2 * P( Z > 1.8257 )

Now P(Z > 1.8257 ) = 1 - P(Z 1.8257) = 1 - 0.9661 = 0.0339

Hence, p-value = 2 * 0.0339 = 0.0678

1) Let = 0.001:
  p-value (0.0678) is greater than (0.001) hence we do not reject the claim, and there is sufficient evidence that over three-quarters of timber pieces of a certain size range contain flaws.

2) Let = 0.01,

p-value (0.0678) is greater than (0.01) hence we do not reject the claim, and there is sufficient evidence that over three-quarters of timber pieces of a certain size range contain flaws.

3) Let = 0.1,

p-value (0.0678) is less than (0.1) hence we reject the claim, and there is no any sufficient evidence that over three-quarters of timber pieces of a certain size range contain flaws.

The result for = 0.1, i.e. for 99% confidence we reject the claim.

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