Elementary statistics

Question

Assume that adults have iq scores that are normally distributed with a mean of u=105 and a standard deviation o=20. Find the probability that a randomly selected adult has an iq less than 137

Answer 1

Hello Chauncey W,

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. 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. If not, let me know if you need further help with substituting the values.

Samantha.

Answer 2

let x = IQ score. want to find P(x < 137).

z = (x - mean)/standard deviation is standardized z-statistic used to find probability from standard normal distribution table .

Convert P(x < 137) to use standardized z-statistic

P(x < 137) = P(z < (137 - 105)/20) = P(z < 1.6) = 0.9452