Mathematics, 30.03.2021 23:10 kataldaine

Suppose that the distribution of IQ's of North Catalina State University's students can be approximated by a normal model with mean 130 and standard deviation 8 points. Also suppose that the distribution of IQ's of Chapel Mountain University's students can be approximated by a normal model with mean 120 and standard deviation 10 points. Required:
What is the probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students?

Answers: 1

Show answers

Answers

0.7517 = 75.17% probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Subtraction of normal variables:

When we subtract normal variables, the mean is the subtraction of the means, while the standard deviation is the square root of the sum of the variances.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

North Catalina:

Population has mean 130 and standard deviation 8 points. Sample of 3. This means that:

$\mu_{N} = 130, s_{N} = \frac{8}{\sqrt{3}} = 4.6188$

Chapel Mountain:

Population has mean 120 and standard deviation 10 points. Sample of 3. This means that:

$\mu_{C} = 120, s_{C} = \frac{10}{\sqrt{3}} = 5.7735$

What is the probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students?

We want that: N - C 5

Distribution N - C:

The mean is:

$\mu = \mu_N - \mu_C = 130 - 120 = 10$

The standard deviation is:

$s = \sqrt{s_N^2+s_C^2} = \sqrt{4.6188^2+5.7735^2} = 7.3937$

This probability is 1 subtracted by the pvalue of Z when X = 5. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{5 - 10}{7.3937}$

Z = -0.68

Z = -0.68 has a pvalue of 0.2483

1 - 0.2483 = 0.7517

0.7517 = 75.17% probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students

Answer from: Quest

the car was traveling 40 mph before it started skidding. ( 2 points ) 17. was the driver were her brakes working at 75% efficiency or better? ( 2 points ) yes 29.

step-by-step explanation:

Answer from: Quest

kayla would of sent 8

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 17:00

Write numerical coefficient of y² in the expression 2x² - 15xy – 7y²

Answers: 1

Answer

Mathematics, 21.06.2019 18:00

The estimated number of people to attend the last bengals game of the season was 2000. the actual number of fans attending was 1200. by what percent did the estimate differ from the actual?

Answers: 3

Answer

Mathematics, 21.06.2019 18:30

Ricky takes out a $31,800 student loan to pay his expenses while he is in college. after graduation, he will begin making payments of $190.66 per month for the next 30 years to pay off the loan. how much more will ricky end up paying for the loan than the original value of $31,800? ricky will end up paying $ more than he borrowed to pay off the loan.

Answers: 1

Answer

Mathematics, 22.06.2019 01:30

Simplify 1/3 12x - 6 a. -2x-1 b.-2x-8 c.-x+4 d.3x+10 e. 4x-2

Answers: 1

Answer

You know the right answer?

Suppose that the distribution of IQ's of North Catalina State University's students can be approxima...

Questions