Hypothesis Testing (4)

Augustus

2012-10-16 2:43 am

4. Bolts are manufactured with a nominal length of 5 cm and it is known from past experience that the variance of the lengths of such bolts is 0.05 cm^2. A random sample of 10 bolts is taken from a box containing a large number of bolts, and their lengths (in cm) are found to be

5.68, 5.13, 5.82, 5.71, 5.36, 5.52, 5.29, 5.77, 5.45, 5.39

(a) Find the 95% confidence interval for the mean length, μ, of bolts in the box, stating clearly any assumptions made in deriving the limits.

(b) Without doing a formal test of a hypothesis, using the result in part (a), discuss whether μ = 5 cm is plausible hypothesis.

(c) Construct a formal test of the null hypothesis H0: μ = 5 cm, against the alternative hypothesis H1: μ ≠ 5 cm. Test at 0.05 level of significance.

(d) Suppose now that σ had not been given, find the 95% confidence interval for the mean length, μ.

回答 (1)

用戶25975

2012-10-16 7:30 pm

✔ 最佳答案

a) Sample μ = 5.512 cm

Hence 95% CI is 5.512 +/- 1.96 x √0.05, between 5.074 cm and 5.950 cm

(Assumption: The length is normally distributed)

b) Not plausible since the lower limit of the CI is still greater than 5 cm.

c) Sample z = (5.512 - 5)/√0.05 = 2.29 > 1.645

Hence at 0.05 level of significance, H1 is accepted.

d) Then σ has to be derived from the sample of 10 bolts (10 samples is still small enough for applying normal distribution)

σ = 0.216

Thus 95% CI is 5.512 +/- 1.96 x 0.216, between 5.089 cm and 5.935 cm

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