Question

1. (20 pts) A major airline is concerned that its proportion of late arrivals has substantially increased in the past month. Historical data shows that on the average 18% of the company airplanes have arrived late. In a random sample of 1240 airplanes, 310 airplanes have late in the past month. Test the claim that the proportion of late arrivals in the past month has increased from 18% at a 0.01 significance level. (a) State the null and alternative hypothesis. (b) Calculate the test statistic. (c) What is the distribution of the test statistic? (d) Draw a graph and determine the critical region.

Answer 1

A.

Hypothesis to be tested is,

H0 : P >=0.18

H0 : proportion of late arrivals of airplanes is greater than or equal to 0.18

V/s

H1 : P < 0.18

H1 : proportion of late arrivals of airplanes is less than 0.18

p^ : sample proportion of late arrivals in the sample

p^ = 310 / 1240 = 0.25

B.

Test statistic under H0 is,

Z = (p^ - P0) /$\sqrt$P0*(1-P0)/n)

Z = (0.25 - 0.18)/$\sqrt$(0.18 * 0.82/1240)

Z = 6.416

C.

The distribution of the test statistic is standard normal distribution (normal distribution with mean 0 and standard deviation 1)

D.

Critical value is given by,

Z$\alpha$ = Z0.01 = 2.32

The critical /rejection region for null hypothesis is,

Z < - Z$\alpha$​​​​​​

Z < - 2.32

(we accept H0(null hypothesis), since Z > - Z$\alpha$​​​​​​)

Critical region t -2.32