Question

A study of the time spent shopping in a supermarket for a marketbasket of 20 specific items showed an approximately uniformdistribution between 20 minutes and 40 minutes. What is theprobability that the shopping time will be What is the probabilitythat the shopping time will be .a. between25 and 30 minutesb. less than 35minutes?c. What are themean and standard deviation of the shopping time?

Answer 1

Answer with explanation:

Probability density function for x which is uniformly distributed [a,b] :

$F(x)=\dfrac{1}{b-a}$

Given : items showed an approximately uniform distribution between 20 minutes and 40 minutes.

Let x be the time spent shopping in a supermarket.

Probability density function for x : $f(x)=\dfrac{1}{40-20}=0.05$

a.

$P(25$

∴The probability that the shopping time will be between 25 and 30 minutes is 0.25.

b.

$P(x$

∴The probability that the shopping time will be  less than 35minutes is 0.5

c.  Mean :  $\mu=\dfrac{a+b}{2}=\dfrac{20+40}{2} =30 \text{ minutes}$

Standard deviations : $\sigma=\sqrt{\dfrac{(b-a)^2}{12}}=\sqrt{\dfrac{(40-20)^2}{12}}\approx5.77\text{ minutes}$

i.e.  the mean and standard deviation of the shopping time are 30 minutes and 5.77 minutes.