A sample size 25 is picked up at random from a population which is normally
distributed with a mean of 100 and variance of 36. Calculate-
(a) P-{ X<99}
(b) P{98X

Answer:

a) P(X < 99) = 0.2033.

b) P(98 < X < 100) = 0.4525

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 z-score 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 p-value, 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.

Mean of 100 and variance of 36.

This means that [tex]\mu = 100, \sigma = \sqrt{36} = 6[/tex]

Sample of 25:

This means that [tex]n = 25, s = \frac{6}{\sqrt{25}} = 1.2[/tex]

(a) P(X<99)

This is the pvalue of Z when X = 99. So

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

By the Central Limit Theorem

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

[tex]Z = \frac{99 - 100}{1.2}[/tex]

[tex]Z = -0.83[/tex]

[tex]Z = -0.83[/tex] has a pvalue of 0.2033. So

P(X < 99) = 0.2033.

b) P(98 < X < 100)

This is the pvalue of Z when X = 100 subtracted by the pvalue of Z when X = 98. So

X = 100

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

[tex]Z = \frac{100 - 100}{1.2}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5

X = 98

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

[tex]Z = \frac{98 - 100}{1.2}[/tex]

[tex]Z = -1.67[/tex]

[tex]Z = -1.67[/tex] has a pvalue of 0.0475

0.5 - 0.0475 = 0.4525

So

P(98 < X < 100) = 0.4525

Which data set could be represented by the box plot shown below?

10
12
14
16
18
20
22
24
26

How is this happening i was only gone for 2 weeks

HELP PLZZZ questions are on the screenshot whats y/4+6=10

HELP ME PLEASE
15 points and brainliest if right

Solve for all values of x:
(x-3)2 = 16
• A X=-7,4
B. X= 4,1
I C. X= 7,11
D. X= 7,-1

Verify the Identity.

tan²xsin²x = tan²x-sin²x

Help me please ASAP I need help

Please help, I need to solve for m. a=f/m

You work and your hourly wage is $12.25 if you work 40 hours this week ,how much money would you make before taxes?

My last question the picture wasnt there soo her eyou go

what is the area of this quadrilateral? plz help

Given the following information, find the length of the missing side. Leave your answer as a simplified radical.

which expressionis 4 times as large as the expression 34 minus 15

The graph of g(x) is a translation of y = RootIndex 3 StartRoot x EndRoot.

On a coordinate plane, a cube root function goes through (negative 6, negative 2), has an inflection point at (4, 0), and goes through (10, 2).

Which equation represents g(x)?

g(x) = RootIndex 3 StartRoot x minus 4 EndRoot
g(x) = RootIndex 3 StartRoot x + 4 EndRoot
g(x) = RootIndex 3 StartRoot x EndRoot + 1.5
g(x) = RootIndex 3 StartRoot x EndRoot minus 1.5

Based on the data im the circle graph if there are 48 total students in the eighth grade, what is the total number of female eighth grade students on academic teams?

A. 48
B. 25
C. 36
D. 12

The ordered pair (10,3) is in quadrant:

Please help this is geometry :)

Solve for x. Round your answer to the nearest tenth.
42
37
X=
units

True or False
Can the expression (2x-2) +2 be read as the sum of two terms?

40 POINTS !! 40 POINTS !!

PLEASE HELP , DONT SKIP !

NO LINKS OR FILES.

1) Which number line below
shows x 4?
a)
1 2 3 4 5 6
b)
0 1 2 3 4 5 6

C)26
O i 2 3 4
D)5
0 1 2
3 4 5
6
A
B
оооо
С
D

Using the photo above, describe what type of symmetry it has.

a. if it has reflection symmetry, draw all lines of symmetry.

b. if it has rotational symmetry, what is all the angle of rotational symmetry (the smallest)

c. if has translational symmetry, what is shape is getting repeated​

What's 1+1=? Pls help

someone please help me answer this!!!

Answer this for me please.

A triangular prism with dimensions is shown below.
5 ft
3 ft
8 ft
What is the volume of the triangular prism in cubic feet?
O A. 120 ft
O B. 20 ft
O C. 40 ft?
O D. 60 ft

A bag contains 2 yellow, 12 red, and 6 green marbles. What is the probability of selecting 1 yellow marble, not replacing it, then selecting a green marble

What is the inverse of y=2x^2-8

Which sign makes this number sentence true?

|-6| __ |6|
A. >
B. <
C. =
D.

Lena purchased a prepaid phone card for $15. Long distance calls cost .17 cents a minute using this card. Lena used her card only once to make a long distance call. If the remaining credit on her card is $7.01, how many minutes did her call last?