Question

In some city there were studied the clocks in shop windows. 5 clocks were found. And the hour hand values were: 1:30, 7:15, 3:15, 0:30, 2:15. Do these data distributed uniformly if this are electronic watches? Do these data distributed uniformly on the 24-hours clock-face? Use Kolmogorov test.

Answer 1

Here we have given 5 observations as follows

1.30, 7.15, 3.15, 0.30, 2.15

Kolmogorov test : The Kolmogorov-Smirnov test (Chakravart, Laha, and Roy, 1967) is used to decide if a sample comes from a population with a specific distribution.

By using r-software have command for kolmogorov test is as follows

ks.test(x, y,alternative=c("two.sided","less","greater"),exact=NULL)

x-a numeric vector of data values.y-either a numeric vector of data values, or a character string naming a cumulative distribution function or an actual cumulative distribution function such as pnorm. Only continuous CDFs are valid.

alternative-indicates the alternative hypothesis and must be one of "two.sided" (default), "less", or "greater".

For your data,all calculations are done in r-software. So I have enclosed all the images of deskstop related to question above.

Here Null hypothesis is

H₀: given observations come from uniform(0,1) vs H₁: given observations does not come from U(0,1).

For question 1 we directly use default command as given in following image.

Result : If p-value is greater than alpha=0.95 then we accept H₀. Here we observe that p-value is less than alpha so here we reject null hypothesis. Means we conclude that given observation does not belong to uniform(0,1)

For question 2 ,we first draw 5 random numbers from uniform(0,24), then we compaire these random numbers with those of observations which we get from clocks in shop.

Result: By observing p-value we conclude that given observations come from uniform(0,24). Means these data distributed uniformly on the 24-hours clock-face.