Question

August, 2018	$250.84
September, 2018	$236.61
October, 2018	$220.70
November, 18	$230.20
December, 2018	$180.32
January 2019	$140.36
February, 2019	$160.22
March, 2019	$128.66
April 2019	$130.12
May 2019	$106.24
June 2019	$176.46
July 2019	$191.45

1. Pick three bills from the last 12 months and change the values into z-scores. What does the z-score tell you about that particular month?
2. Between what two values would be considered a normal bill? Remember, being within 2 Standard Deviations is considered normal.

Answer 1

We are given the bill charges for the last 12 months. We will first calculate the mean and standard deviation of the last 12 months bill.

Mean of the last 12 months bill, x̄ = $(250.84 + 236.61+----------------------------------------------+ 176.46 + 191.45)/12

x̄ = $179.35

Standard deviation of the last 12 months bill, σ = $√{(250.84 - 179.35)² + (236.61 - 179.35)² +----------------------+ (176.46 - 179.35)² + (191.45 - 179.35)²/11

σ = $47.74

I will pick October, 2018, January, 2019 and May, 2019 to calculate the z-scores for these months.

Z- score for October 2018:

Z- score = (x̄ - µ)/σ

= (179.35 - 220.70)/47.74

= -41.35/47.74

= -0.87

Therefore, Z- score for October 2018 is -0.87.

Z- score for January 2019:

Z- score = (x̄ - µ)/σ

= (179.35 - 140.36)/47.74

= 38.99/47.74

= 0.82

Therefore, Z- score for January 2019 is 0.82.

Z- score for May 2019:

Z- score = (x̄ - µ)/σ

= (179.35 - 106.24)/47.74

= 73.11/47.74

= 1.53

Therefore, Z- score for May 2019 is 1.53.

The value of the z-score tells you how many standard deviations you are away from the mean.

The Z- score for October 2018 is -0.87 which tells us that it is 1 standard deviation below the mean. This means that the Z- score for October 2018 is less than the average bill score.

The Z- score for January 2019 is 0.82 which tells us that it is 1 standard deviation above the mean. This means that the Z- score for October 2018 is more than the average bill score.

The  Z- score for May 2019 is 1.53 which tells us that it is approximately 1.5 standard deviation above the mean. This means that the Z- score for October 2018 is much more than the average bill score.

A bill is considered normal if it is within two Standard Deviations of the average bill.

Let us find the bill under two Standard Deviations.

Two Standard Deviations below the mean bill:

Bill = x̄ - 2*σ

= $(179.35 - 2*47.74)

= $(179.35 - 95.47)

= $83.88

Thus, a bill below the two Standard Deviations of the mean bill is $83.88.

Two Standard Deviations above the mean bill:

Bill = x̄ - 2*σ

= $(179.35 + 2*47.74)

= $(179.35 + 95.47)

= $274.82

Thus, a bill above the two Standard Deviations of the mean bill is $274.82.

In conclusion, a bill would be considered normal if it is between $83.88 and $274.82.