Question

Solve it by R

Over the past 5 years, the mean time for a warehouse to fill a buyer’s order has been 25 minutes. Officials of the company believe that the length of time has increased recently, either due to a change in the workforce or due to a change in customer purchasing policies. The processing time (in minutes) was recorded for a random sample of 15 orders processed over the past month. 28 25 27 31 10 26 30 15 55 12 24 32 28 42 38

(1) Check the normality of the data.

(2) Please test the research hypothesis at the significance level α = 0.05.

Answer 1

Solution :- Let be the population mean.The hypothesis want to test is given as

  H0:=25 V/s H1:

Let be the level of significance.

(1)-:

The hypothesis for normality is

H0: Given data follow normality. v/s

H1: Given data does not follow normality.

Normality of the data by using R is

R-output:

> x=c(28,25,27,31,10,26,30,15,55,12,24,32,28,42,38)
> shapiro.test(x)

Shapiro-Wilk normality test

data: x
W = 0.94167, p-value = 0.4038

Here p-value=0.4038 >0.05 we accept  H0  at 5% level of significance.

Conclusion: From R output and p-value, Given data follows normality.

(2)-:

Since from (1) , data follows normal distribution.Here population variance is unknown

so we use t-test for population mean equal to specified value.

R-output:-

> x=c(28,25,27,31,10,26,30,15,55,12,24,32,28,42,38)
> t.test(x,mu=25,alternative="greater")

One Sample t-test

data: x
t = 1.0833, df = 14, p-value = 0.1485
alternative hypothesis: true mean is greater than 25
95 percent confidence interval:
22.99721 Inf
sample estimates:
mean of x
28.2

Here p-value=0.1458 is greater than 5% level of significance, so we accept H0  . Also the 95% confidence interval

( 22.99721,)  includes value of =25. Hence we accept H0.

Conclusion: There is sufficient evidence to prove that mean length of time does not increased recently. Mean length of time is equal to 25 minutes.