Data Management Probability, Standard Deviation, and Normal Distribution Question:

Question

Hi, how would i get started on solving this question?

The average monthly rainfall in Southern California is 26 mm. If rainfall data over the year is normally distributed with a standard deviation of 3.8 mm. Calculate the probability that it will rain less than 30 mm. Show all your work and include a sketch for the z-scores.

Answer 1

Hello Rahman,

This is a simple problem where you pretty much just substitute the values in the Z-Score formula to get the answer. Here is really good and simplified explanation provided by fellow tutor Kiran B to a similar question. https://www.wyzant.com/resources/answers/772059/finding-probability-with-means-and-standard-deviation

You need to know two things to answer this question. First, know how to calculate standard score or z-score and then know how to find probability from the z-score.

Formula for calculating the standard score or z score:

z = x-μ/σ, where:

z is the standard score

x is the raw score 

μ is the population mean

σ is the population standard deviation

In this question: μ = 82 and σ = 9

a) The score is less than 77 

Let’s calculate the z score, for x = 77 and then find the probability for x less than 77 

z = 77-82/9 = -0.56

Now, let’s look at the normal standard distribution table to find the probability. 

 You can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 

The z value upto the first decimal point is in rows. The second decimal point is in columns. 

So, for z = -0.56, look at the row -0.5 and column 0.06 to find the probability.

The probability is 0.28774 

Therefore, the answer is 0.28774

b) The score is greater than 65 

Let’s calculate the z score, for x = 65 and then find the probability for x greater than 65

z = 65-82/9 = -1.89

Now, let’s look at the normal standard distribution table to find the probability. 

 Again, you can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 

The z value upto the first decimal point is in rows. The second decimal point is in columns. 

So, for z = -1.89, look at the row -1.8 and column 0.09 to find the probability.

The probability is 0.02938. However, we need to find for x > 65. To find the probability for greater than x, we need to subtract the probability from 1

1-0.02938 = 0.97062

Therefore, the answer is 0.97062

c) The score is between 65 and 80

In this case, we need to find z and probability twice. Once, for x = 65, and then for x = 80.

From above we already know that for x = 65, the z score = -1.89 and probability = 0.02938.

Let’s find the z and probability for x = 80.

z = 80-82/9 = -0.22

Now, let’s look at the normal standard distribution table to find the probability. 

 Again, you can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 

The z value upto the first decimal point is in rows. The second decimal point is in columns. 

So, for z = -0.22, look at the row -0.2 and column 0.02 to find the probability.

The probability is 0.41294.

So, the probability for x = 65 is 0.02938 and the probability for x=80 is 0.41294.

Since we need to calculate the probability for the score between 65 and 80, we subtract the two probabilities i.e.,  0.41294- 0.02938 = 0.38356.

Therefore the answer is 0.38356.

Let me know if you're able to solve your query using the above explanation. Substitute your values in the above z-score equation and see if you can figure it out. If not, let me know if you need further help with substituting the values.

Samantha.

Answer 2

Hello Rahman,

Draw a normal curve with 26 in the middle. Put 30 to the right of center and shade "less than 30" so you can see it is over 50% of the area. To find the area, you can use a statistical calculator (normalcdf) , computer software, or find the z-score for 30 and use a table.

To find the z-score, use z = (Xscore - mean)/standard deviation.

See if you can work it out from there.

Tom S.