STA 2023 FIU The Central Limit Theorem and Sampling Distributions Math Lab Task Please help me solve my lab 5 questions. Excel Labs attached with pdf instr

STA 2023 FIU The Central Limit Theorem and Sampling Distributions Math Lab Task Please help me solve my lab 5 questions. Excel Labs attached with pdf instructions. Will need answers to questions in a word doc. LAB 5: The Central Limit Theorem and Sampling Distributions
Learning Objectives
• Outline the differences between continuous and discrete random variables.
• Determine event probabilities for a normal random variable, given values of µ and
σ.
• Compute the mean and standard error of the sampling distribution of X̄ for sample
size N, µ and σ .
• State the Central Limit Theorem.
• Assess the precision of X̄ and p̂ as point estimates of µ and the binomial parameter
p.
Key Concepts
Continuous Random Variables
You plan to participate in an environmental conservation project, a worthy cause! Your
job is to collect a sample of frogs from a marsh and record each frog’s weight in grams.
Let the random variable X represent an individual frog’s weight in grams. How many
possible frog weights fill up the sample space? The average frog weighs 23 grams,
although an individual frog may weigh 23.267 grams or 22.559 grams. There is a large
(even infinite) number of possible frog weights! When the sample space for a random
variable X is infinite and uncountable, defined by a continuous interval of real numbers,
then X is a continuous random variable.
For a finite sample space, the probability of an event equals the sum of all probabilities
associated with the event (The Classical Method). For a continuous random variable
X and its sample space, we need a new approach! Not only is the sample space of X
infinite and uncountable, events such as {X > 22 grams} also contain an infinite and
uncountable number of sample points.
Probability Density Function (PDF)
A probability density function (PDF) for a continuous random variable is a smooth function that circumscribes an area equal to one. A possible PDF for X is depicted here:
Probability Density Function (PDF) of X: Frog’s Weight in Grams
The PDF determines event probabilities for a continuous random variable X. The probability P(X > 22 grams) equals the area under the PDF curve where {X > 22 grams}.
Probability Density Function (PDF) of X: Frog’s Weight in Grams
As illustrated above, P(X > 22 grams) = 0.841.
Normal Random Variables
A normal random variable X is a continuous random variable having the following general PDF:
Probability Density Function (PDF) of a Normal Random Variable
The mean (µ) defines the center of the normal PDF and the standard deviation (σ )
defines its spread. The areas under the normal PDF curve follow the empirical rule,
which states:
68% of all values of X lie within one standard deviation (1σ ) of µ.
95% of all values of X lie within two standard deviations (2σ ) of µ.
99.7% of all values of X lie within three standard deviations (3σ ) of µ.
X −µ
A Z transformation of any normal random variable X produces Z = σ , so that Z is a
normal random variable, having µ=0 and σ =1. A Z-table reports areas under the PDF
curve of Z. You will want to have a Z-table handy in order to compute event probabilities
for a normal random variable X.
Let the normal random variable X having µ = 23 and σ = 1 represent an individual
frog’s weight in grams. The following event probabilities can be computed with the aid
of a Z-table:
a) P(X > 22 grams)
b) P(X > 25 grams)
c) P( 22 grams 22 grams)
P(X > 22 grams) corresponds with P(Z > −1). Based on a Z-table, this probability
equals 0.841.
b) PDF of a Normal Random Variable with µ=23 and σ =1: P(X > 25 grams)
P(X > 25 grams) corresponds with P(Z > +2). Based on a Z-table, this probability
equals 0.023.
c) PDF of a Normal Random Variable with µ=23 and σ =1: P( 22 grams