Question

The mass of garbage put out for curbside pickup, per household, in a certain community, is reported to have a mean of 340 kg/year and a standard deviation of 85 kg/year. Based on this information:
a) Assume that the population is very large. For a sample of 200 households from this community, calculate, to the nearest %, the probability that the sample mean will be between 335 and 345 kg/year.
b) Redo Part (a), for a sample size of 1,000 households instead of 200.
c) Redo Part (a) (ie, n = 200), but this time assume a population of 500 households.
d) Assuming again that the population is very large, let us now suppose that 4 consecutive samples of 200 households are taken, and each one of them has a sample mean less than 330 kg/year. Calculate the probability of obtaining this result.
e) Based on your answer for Part (d), is there reason to believe that the mean weekly garbage output per household is not actually around 340 kg/year? If so, in which direction has the mean garbage output shifted? Explain your answers using your previous calculations for this question, plus your knowledge of statistics fundamentals

Answer 1

a)

Given:

Mean	340
SD	85
Sample size, n	200

The probability is obtained from standard normal distribution table by calculating the z score as follow,

Converting to standard normal distribution,

Now the probability is obtained from standard normal distribution table for the calculated z score, (in excel use formula =NORM.S.DIST(z,TRUE))

b)

Replacing the sample size n = 200 by n = 1000.

c)

Replacing the sample size n = 200 by n = 500.

d)

The probability of sample mean is less than 330 in each draw is independent event. Hence using the law of independent of probabilities such that, P(A and B) = P(A).P(B)

Now, probability of sample mean is less than 330 in single draw is obtained by same procedure as done above,

using the law of independent of probabilities, for 4 independent draw,

e)

Since the probability is very low, there is no reason to believe that the mean weekly garbage output per household is not actually around 340 kg/year