finding probability with means and standard deviation

Question

Assume that a set of test scores is normally distributed with a mean of 82

and a standard deviation of 9.

Find probabilities that

a) The score is less than 77.

b) The score is greater than 65.

c) The score is between 65 and 80.

Answer 1

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:

1. z is the standard score
2. x is the raw score 
3. μ is the population mean
4. 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 

1. z = 77-82/9 = -0.56

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

1. You can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 
2. The z value upto the first decimal point is in rows. The second decimal point is in columns. 
3. So, for z = -0.56, look at the row -0.5 and column 0.06 to find the probability.
4. 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

1. z = 65-82/9 = -1.89

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

1. Again, you can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 
2. The z value upto the first decimal point is in rows. The second decimal point is in columns. 
3. 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. 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.

1. z = 80-82/9 = -0.22

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

1. Again, you can find the probability from z-score from a table like shown in this link: https://www.ztable.net/ 
2. The z value upto the first decimal point is in rows. The second decimal point is in columns. 
3. 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.

Answer 2

z = (77 - 82) / 9 = -5/9 = -0.55

P(z < -0.55) = 0.2912

z = (65 - 82) / 9 = -17/9 = -1.89

P(z > -1.89) = 1 - P(z ≤ -1.89) = 1 - 0.0294 = 0.9706

z = (80 - 82) / 9 = -2/9 = -0.22

P(-1.89 < z < -0.22) = P(z < -0.22) - P(z < -1.89) = 0.4129 - 0.0294 = 0.3838

a.) The probability that a test score is less than 77 is 0.2912.

b.) The probability that a test score is more than 65 is 0.9706.

c.) The probability that a test score is between 65 and 80 is 0.3838.