Statistics from a college's climate center indicate that the city the college is in gets an average of 35.4" of rain each year, with a standard deviation of 4.1". Assume that a Normal model applies. Complete parts a through d below.
Question content area bottom
Part 1
a) During what percentage of years does this city get more than 39" of rain?
The percentage of years with more than 39" of rain is

enter your response here%.
(Round to two decimal places as needed.)
Part 2
b) Less than how much rain falls in the driest 25% of all years?
The driest 25% of all years get at most

enter your response here" of rain.
(Round to one decimal place as needed.)
Part 3
c) A university student is in this city for 7 years. Let y overbar represent the mean amount of rain for those 7 years. Describe the sampling distribution model of this sample mean, y overbar. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A.
A Normal model with mean

enter your response here" and standard deviation

enter your response here"
(Round to two decimal places as needed.)
B.
A Binomial model with

enter your response here trials and a probability of success of

enter your response here
(Type integers or decimals rounded to two decimal places as needed.)
C.
There is no model that fits this distribution.
Part 4
d) What's the probability that those 7 years average less than 30" of rain?
The probability that those 7 years average less than 30" of rain is

enter your response here.
(Round to four decimal places as needed.)

To solve these problems, we will use properties of the Normal distribution. Given the average annual rainfall of 35.4 inches with a standard deviation of 4.1 inches, we'll approach each question using the Normal model.

Part 1: Percentage of Years with More Than 39 Inches of Rain

To find this, we first calculate the z-score for 39 inches of rain and then use the standard normal distribution to find the corresponding percentage.

=

−

=

39

−

35.4

4.1

z=

σ

X−μ

=

4.1

39−35.4

Let's calculate the z-score:

=

39

−

35.4

4.1

≈

0.878

z=

4.1

39−35.4

≈0.878

Next, we find the percentage of years with more than 39 inches of rain. We'll use standard normal distribution tables or a calculator for this.

Part 2: Rainfall for the Driest 25% of All Years

To find the rainfall amount that corresponds to the driest 25% of years, we find the 25th percentile of the normal distribution.

For the 25th percentile (bottom 25%), the corresponding z-score is approximately -0.674.

=

+

=

35.4

+

(

−

0.674

)

(

4.1

)

X=μ+zσ=35.4+(−0.674)(4.1)

Let's calculate this:

=

35.4

−

2.7634

≈

32.6

X=35.4−2.7634≈32.6

Part 3: Sampling Distribution Model of the Sample Mean for 7 Years

For a sample mean of 7 years, the distribution of the sample mean will also be normal due to the Central Limit Theorem. The mean of the sampling distribution will be the same as the population mean, and the standard deviation will be the population standard deviation divided by the square root of the sample size.

Mean:

Mean

=

35.4

Mean=μ=35.4

Standard Deviation:

Standard Deviation

=

4.1

7

Standard Deviation=

n

σ

=

7

4.1

Part 4: Probability that the 7-Year Average is Less Than 30 Inches of Rain

We find the z-score for 30 inches of rain using the sample mean distribution.

=

30

−

=

30

−

35.4

4.1

7

z=

n

σ

30−μ

=

7

4.1

30−35.4

Let's calculate these values.

It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can provide you with the steps and formulas to complete these calculations:

Part 1: Percentage of Years with More Than 39 Inches of Rain

Calculate the z-score:

=

39

−

35.4

4.1

z=

4.1

39−35.4

Then, find the percentage using a standard normal distribution table or calculator.

Part 2: Rainfall for the Driest 25% of All Years

Use the 25th percentile z-score (-0.674) and calculate the corresponding rainfall:

=

35.4

+

(

−

0.674

)

×

4.1

X=35.4+(−0.674)×4.1

Part 3: Sampling Distribution Model for 7 Years

Calculate the standard deviation for the sampling distribution:

Standard Deviation

=

4.1

7

Standard Deviation=

7

4.1

The mean remains 35.4.

Part 4: Probability of Less Than 30 Inches Over 7 Years

Calculate the z-score:

=

30

−

35.4

4.1

7

z=

7

4.1

30−35.4

Then, find the probability using the standard normal distribution