Suppose a certain manufacturer of building components claims that at least 95% of the building
components they produce conform to specifications. An examination of 500 various components
supplied by the firm revealed 40 faulty ones. Does this result provide sufficient evidence to reject the
manufacture’s claim at 5% decision level?
To test whether the manufacturer's claim can be rejected at a 5% decision level, we can conduct a one-sample proportion hypothesis test.
Null hypothesis (H0): p = 0.95 (proportion of conforming components)
Alternative hypothesis (H1): p < 0.95 (proportion of conforming components is less than claimed)
The test statistic for this hypothesis test is a z-score, which is calculated using the formula:
z = (p̂ - p) / sqrt(p * (1 - p) / n)
Where:
p̂ is the sample proportion of faulty components (40/500 = 0.08)
p is the claimed proportion of conforming components (0.95)
n is the sample size (500)
Using these values, we can calculate the z-score:
z = (0.08 - 0.95) / sqrt(0.95 * (1 - 0.95) / 500)
= (-0.87) / sqrt(0.0475 / 500)
≈ -14.55
At a 5% decision level, the critical z-value for a one-tailed test is approximately -1.645 (obtained from the standard normal distribution table).
Since the calculated z-score (-14.55) is much smaller than the critical z-value (-1.645), we can reject the null hypothesis. There is sufficient evidence to conclude that the manufacturer's claim of at least 95% conforming components is not supported by the data.