Question

An engineer measured the diameter of bearings in cm. 10.2 16.9 13.3 12.5 12.1 10.0 10.9 13.5 16.6 11.1 12.0 16.2 12.3 11.8 13.0 (it is okay to get the mean and sd by Ror STAT mode of your calculator) It is believed that the diameters of bearings form a normal distribution. a. Find an efficient interval that 90% of the data would lie, b Find an efficient interval that the population mean diameter of bearings would lle with 90% confidence c Test whether the population mean diameter of bearings is 14 at a 50.05 by the critical value approach. dTest if population mean diameter of bearings is less than 14 at a 50.1 by the p-value approach (If there are the same steps or quantities as in (c), you can skip them by simply referring to them or copying the result.) e. Can we derived the conclusion of the test in (d) based on the interval derived in (b)? If yes, explain explicitly how we can draw the same conclusion based on the interval in (b). If not, explain explicitly why we can't. f. Without any further computations, make a conclusion of test if the population mean diameter of bearings is 14 at < =0.1.

Answer 1

x = 12.83

s = 2.19

(a) t-critical value = 1.76

= x ± t*(s/√n)

= 12.83 ± 1.76*(2.19/√15)

= 11.8307, 13.8226

(b) We are 90% confident that the population mean diameter of bearings is between 11.8307 and 13.8226.

(c) The hypothesis being tested is:

H₀: µ = 14

H_a: µ ≠ 14

The test statistic, t = (x - µ)/s/√n

t = (12.83 - 14)/2.19/√15 = -2.075

The critical value is 2.145.

Since 2.075 > 2.145, we fail to reject the null hypothesis.

Therefore, we cannot conclude that µ ≠ 14.

(d) The hypothesis being tested is:

H₀: µ = 14

H_a: µ < 14

The test statistic, t = (x - µ)/s/√n

t = (12.83 - 14)/2.19/√15 = -2.075

The p-value is 0.0285.

Since the p-value (0.0285) is less than the significance level (0.10), we can reject the null hypothesis.

Therefore, we can conclude that µ < 14.

(e) Since 14 is above the confidence interval's upper limit, we can say that µ < 14.

(f) We cannot conclude that µ = 14.