The overhead reach distances of adult females are normally distributed with a mean of 200 cm and a standard deviation of 8.3 cm. a. Find the probability that an individual distance is greater than 210.00 cm. b. Find the probability that the mean for 20 randomly selected distances is greater than 197.80 cm c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

Answer:

a) 0.1151 = 11.51% probability that an individual distance is greater than 210.00 cm.

b) 0.883 = 88.3% probability that the mean for 20 randomly selected distances is greater than 197.80 cm.

c) The distribution of the population is normal, which means that the sample size does not need to exceed 30.

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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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 pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

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

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

The overhead reach distances of adult females are normally distributed with a mean of 200 cm and a standard deviation of 8.3 cm.

This means that [tex]\mu = 200, \sigma = 8.3[/tex]

a. Find the probability that an individual distance is greater than 210.00 cm.

This is 1 subtracted by the pvalue of Z when X = 210. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{210 - 200}{8.3}[/tex]

[tex]Z = 1.2[/tex]

[tex]Z = 1.2[/tex] has a pvalue of 0.8849

1 - 0.8849 = 0.1151

0.1151 = 11.51% probability that an individual distance is greater than 210.00 cm.

b. Find the probability that the mean for 20 randomly selected distances is greater than 197.80 cm

Now [tex]n = 20, s = \frac{8.3}{\sqrt{20}} = 1.8559[/tex]

This probability is 1 subtracted by the pvalue of Z when X = 197.8. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{197.8 - 200}{1.8559}[/tex]

[tex]Z = -1.19[/tex]

[tex]Z = -1.19[/tex] has a pvalue of 0.117

1 - 0.117 = 0.883

0.883 = 88.3% probability that the mean for 20 randomly selected distances is greater than 197.80 cm.

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

The distribution of the population is normal, which means that the sample size does not need to exceed 30.

According to Cavaleri's Principle, if a cone and a pyramid have the same height, then

Tell me the Definition of each Word listed (math)

Determine whether the following sequence is geometric. If so, find the common ratio.
1, -5, 25, - 125, ...

A customer at least lunches can choose a protein from chicken beef or shrimp the customer can also choose from right or brown rice which diagram shows all the possible combinations of one type of Protane and one type of rice

Which sum represents the partial fraction decomposition of 8x^ 4 + 3x^3- 115x^2- 39x+200/ 3x (x^2-10)^2

What is the percentage of data between 90 and 140?

FREE BRAIN LIST NEED HELP ASAP

What is the solution to 4x+63 ≤ 18?​

What is the value of y in the following equation? -10y + 4(3y – 8) = -64

Help me me please now

Can someone help meee

Glados drew a triangle with the base of 9.6 centimeters. It's area was 30.72 square centimeters. What was it's height (nearest tenth)​

6) A garden is 90m long and 50m wide. A path of 5m width is built outside and around it.
Find
(1) the area of the path
(11) the cost of cementing the path at the rate of Rs 250 per mº​

Helpppp PLZ 20 POINTS!!!!! More then one answer

Can someone please help meeeee

If you were to calculate the average rate of change of a function over a variety of intervals and you found that they were all the same, what conclusion could you draw and why

Find x. Please and thank you!

Simplify 5 xa x 5 x a​

I cut a pizza in half. Then I cut each half into 3 equal pieces and each piece into 2 equal slices. I sold 6 slices. What fraction of the pizza did I sell?​

Five forces are acting on a point P. They are 60N at 90 degrees, 40N at 0 degrees, 80N at 270 degrees, 40N at 180 degrees, and 50 N at 60 degrees. What is the magnitude and direction of the vector that would produce equilibrium at point P?

A circular pond has a radius of 6m. Calculate the circumference of the pond. Give your answer rounded to the nearest tenth.

Find the value of, x/2 + x/3 + x/6

In ΔMNO, the measure of ∠O=90°, the measure of ∠M=66°, and NO = 60 feet. Find the length of MN to the nearest tenth of a foot.

3 (4x - 2) - 5x = 6 - 7 (-x -2)

Add and Subtract Linear Expressions


1. Subtract m + 8 ​​from​ 5m + 11.

Enter your answer in the box.


2. Add the two expressions.

−4.2x+3 and 2.5x−6

Enter your answer in the box.

The sum of a number and twice a second number is 29. The second number is ten less than three times the first number. What are the two numbers?

Also, write the two equations.

L(-5, -1)
L'd
M(-3, 1). -
M'(
N(-1,8)
Nd

What the degrees measure for 5/9 of a circle

WILL GIVE BRAINLIEST. NEED URGENTLY!

Use the FOIL method to find the product below.
(x + 3)(x2 - 6x)
A. x3 - 9x2 - 18
B. X8 - 9x2 - 18x
c. x3 - 3x2 - 18x
D. x2 – 3x2 - 18

What is the answer for this 5c + 9 -2c -7