## Problem Description:

You are given a set of questions related to statistical analysis assignment and probability. Each question requires you to apply the Central Limit Theorem and various statistical methods to find probabilities and percentiles. Below are the solutions to each question.

## Solution

### Question 1

**μ=30
**

**σ=20
**

**n=100
**

From the Central Limit Theorem, if a sample size is sufficiently large, n≥30, then the sample mean is approximately normally distributed with mean, μ_x ̿ =μ and standard deviation,

**σ_x ̿ =σ/√n
**

Therefore, the sampling distribution of x ̅, has mean μ_x ̿ =30 and standard deviation,

**σ_x ̿ =20/√100=2
**

### Question 2

**n=150
**

**μ=240
**

**σ=28
**

**P(x ̿>4)=P(z>(239-240)/(28/√150))
**

**=P(z>(-1)/2.28619)
**

**=P(z>-0.4374)
**

**=1-P(z<-0.4374)
**

**=1-0.3309
**

**=0.6691
**

### Question 2b

38th Percentile of the population

**38/100=ϕ((x ̿-240)/(28/√150))
**

**⟹ϕ^(-1) (0.38)=(x-240)/2.28619
**

**⟹-0.305×2.28619=x-240
**

**∴-0.6984=x-240
**

**x=240-0.6984
**

**=239.30
**

### Question 2c

**P(239**

**=P(-0.4374**

**=P(z<0.4374)-P(z<-0.4374)
**

**=0.6691-0.3309
**

**=0.3382**

### Question 2d

**P(x ̿>242)=P(z>(242-240)/(28/√150))
**

**=P(z>2/2.28619)
**

**=P(z>0.8748)
**

**=1-P(z<0.8748)
**

**=0.1908**

It is not unusual for the sample mean to be more than 242 since the calculated probability = 0.1908 is within the bounds of probability value; 0≤P≤1.

### Question 2e

**P(238**

**=P(-0.8748**

**=P(z<0)-P(z<-0.8748)
**

**=0.5-0.1908
**

**=0.3092**

### Question 3a

μ=2.25

σ=1.3

n=90

P(x ̿>2)=P(z>(2-2.25)/(1.3/√90))

=P(z>-1.8244)

=1-P(z<-1.8244)

=1-0.034

=0.966

### Question 3b

P(2.3

=P(0.3649

=P(z<6.9327)-P(z<0.3649)

=0.3576

### Question 3c

20th Percentile of the population

20/100=ϕ((x ̿-2.25)/(1.3/√90))

⟹ϕ^(-1) (0.2)=(x-2.25)/0.1370

⟹-0.8416×0.1370=x-2.25

∴x=2.25-0.08870

=2.16

### Question 3d

P(x ̿<2)=P(z<(2-2.25)/(1.3/√90))

=P(z<-1.8244)

=0.034

It would not be unusual for the sample mean to be less than 2 since the calculated probability =0.034 is within the bounds of probability value; 0≤P≤1.

### Question 3e

P(x<2)=P(z<(2-2.25)/1.3)

=P(z<-0.1923)

=P(-0.1923)

=0.4238