a study of the time spent shopping in a supermarket for a market basket of 20 specific items showed an approximately uniform distribution between 20 minutes and 40 minutes. What is the probability that the shopping time will be

a) between 25 and 30 minutes?
b) less than 35 minutes
c) what are the mean and standard deviation of the shopping time

If it is a uniform (not normal) distribution, the range = 40 -20 = 20

a. 5/20
b. 15/20
c. Mean = 30
(I don't know how one could determine SD in this case.)

wdf

P(X<35)= (20<X<35) =15/20

P=Varx=((b-a)^2)/12 = ((40-20)^2/12=33.333? may be

a) Ah, let me calculate that for you. Since the distribution is uniform, the probability of the shopping time being between 25 and 30 minutes is the same as the probability of it being between 20 and 30 minutes minus the probability of it being between 20 and 25 minutes. So, (30 - 20) / (40 - 20) - (25 - 20) / (40 - 20) gives us the answer. Just give me a second... *Calculating noises* Ah, it seems like the probability is 1/4 or 0.25.

b) To determine the probability of the shopping time being less than 35 minutes, we need to find the proportion of the total range (20-40 minutes) that is less than 35 minutes. So, (35 - 20) / (40 - 20) equals the answer. Let me get my calculator... *tap tap tap* And voila! The probability is 1/4 or 0.25.

c) Now, for the mean and standard deviation of the shopping time. Since the distribution is uniform, the mean is simply the average of the minimum and maximum time values. So, (20 + 40) / 2 equals the mean. Ah, look at that! The mean is 30 minutes.

As for the standard deviation, it can be calculated using the formula (max - min) / √12. Plug in the values, (40 - 20) / √12, and after some number crunching... *mathy sounds* Ah, the standard deviation is approximately 4.08 minutes.

Have fun shopping and remember, laughter is the best discount!

To find the probabilities and statistics for the given scenario, we can use the properties of a uniform distribution. The uniform distribution is characterized by a continuous probability density function, where all values between two given endpoints are equally likely.

a) To find the probability that the shopping time is between 25 and 30 minutes, we need to calculate the proportion of the total distribution that falls within this interval. Since the distribution is uniform, the probability is equal to the width of the interval divided by the total width of the distribution.

Width of the interval = 30 minutes - 25 minutes = 5 minutes
Total width of the distribution = 40 minutes - 20 minutes = 20 minutes

Probability = Width of the interval / Total width of the distribution
Probability = 5 minutes / 20 minutes = 0.25 or 25%

Therefore, the probability that the shopping time will be between 25 and 30 minutes is 25%.

b) To find the probability that the shopping time is less than 35 minutes, we need to calculate the proportion of the total distribution that falls below 35 minutes. Again, since the distribution is uniform, the probability is equal to the width of the interval divided by the total width of the distribution.

Width of the interval = 35 minutes - 20 minutes = 15 minutes

Probability = Width of the interval / Total width of the distribution
Probability = 15 minutes / 20 minutes = 0.75 or 75%

Therefore, the probability that the shopping time will be less than 35 minutes is 75%.

c) The mean and standard deviation of a uniform distribution can be calculated using the following formulas:

Mean = (a + b) / 2, where a is the lower endpoint and b is the upper endpoint of the distribution.
Mean = (20 minutes + 40 minutes) / 2 = 30 minutes

Standard Deviation = (b - a) / √12, where a is the lower endpoint and b is the upper endpoint of the distribution.
Standard Deviation = (40 minutes - 20 minutes) / √12 = 20 minutes / √12 ≈ 5.77 minutes

Therefore, the mean shopping time is 30 minutes, and the standard deviation is approximately 5.77 minutes.