MATH 110 CHAPTER 7, 8, 9 QUESTIONS
L
WITH COMPLETE SOLUTION
Probability Density Curve - the curve used to describe the distribution of a continuous random variable
Normal Curve - the bell-shaped curve that describes the distribution of many physical and psychological
attributes. Most scores fall near the average, and fewer and fewer scores lie near the extremes
Z-Score - a measure of how many standard deviations you are away from the norm (average or mean)
Example: A woman whose height is x=67 inches from a normal population with mean μ=64 and σ=3
inches.
What is the Z-score? - The Z-score is z = (x - μ)/σ = (67-64)/3 = 1
z score formula - z = (x - μ)/σ
finding the area under the normal curve - normalcdf(lower, upper, μ, σ)
To get it: 2nd+Var -> 2: norm
Example: Find the area to the left of z = 1.26 in a normal curve - normalcdf(-1E99, 1.26, 0, 1)
Example: A study reported that the length of pregnancy from conception to birth is approximately
normally distributed with mean μ = 272 days and standard deviation σ = 9 days. What proportion of
pregnancies last less than 259 days? - Step 1: Find the Z-Score. (259-272)/9 = -1.444
Step 2: Use the Z-Score to find area left of Z = -1.444
normalcdf(-1E99, -1.444, 0, 1)
Finding X from z-score - invNorm(Area to the left, μ, σ)
To get to it: 2nd+Vars -> 3:invNorm
Mensa is an organization whose members possess IQs in the top 2% of the population. If the IQs are
normally distributed with a mean of 100 and a standard deviation of 15, what is the minimum IQ
necessary for admission? - First: Find the z-score that corresponds to the given area using invNorm.
, z = invNorm(0.98, 0, 1)
x = μ + z * σ = 100 + 2.0537(15)
x = 130.8
x ≈ 131
mean of sampling distribution of x̄ - μx̄ = μ
The mean of the sampling distribution is denoted by μx̄ and is equal to the mean of the population μ.
standard deviation of sampling distribution of x̄ - σx̄ = σ/√n
The standard deviation of the sampling distribution (Standard Error) is denoted by σx̄ and equals the
standard deviation of the population divided by the square root of the sample size.
Sampling distribution of a skewed population - if a population is skewed, a larger sample size is necessary
for the sampling distribution of x̄ to be approximately normal.
The Central Limit Theorem - The theory that, as sample size increases, the distribution of sample means
of size n, randomly selected, approaches a normal distribution.
In practice, a sample size of n>30 is enough.
Example: Based on data from the U.S. Census, the mean age of college students in 2011 was μ = 25
years, with a standard deviation of σ = 9.5 years. A simple random sample of 125 students is drawn.
What is the probability that the sample mean age of the students is greater than 26 years? - Step 1:
Confirm the assumptions.
n=135>30
Thus, we may use the normal curve.
Step 2: Then, compute μx̄ and σx̄.
μx̄ = μ = 25
L
WITH COMPLETE SOLUTION
Probability Density Curve - the curve used to describe the distribution of a continuous random variable
Normal Curve - the bell-shaped curve that describes the distribution of many physical and psychological
attributes. Most scores fall near the average, and fewer and fewer scores lie near the extremes
Z-Score - a measure of how many standard deviations you are away from the norm (average or mean)
Example: A woman whose height is x=67 inches from a normal population with mean μ=64 and σ=3
inches.
What is the Z-score? - The Z-score is z = (x - μ)/σ = (67-64)/3 = 1
z score formula - z = (x - μ)/σ
finding the area under the normal curve - normalcdf(lower, upper, μ, σ)
To get it: 2nd+Var -> 2: norm
Example: Find the area to the left of z = 1.26 in a normal curve - normalcdf(-1E99, 1.26, 0, 1)
Example: A study reported that the length of pregnancy from conception to birth is approximately
normally distributed with mean μ = 272 days and standard deviation σ = 9 days. What proportion of
pregnancies last less than 259 days? - Step 1: Find the Z-Score. (259-272)/9 = -1.444
Step 2: Use the Z-Score to find area left of Z = -1.444
normalcdf(-1E99, -1.444, 0, 1)
Finding X from z-score - invNorm(Area to the left, μ, σ)
To get to it: 2nd+Vars -> 3:invNorm
Mensa is an organization whose members possess IQs in the top 2% of the population. If the IQs are
normally distributed with a mean of 100 and a standard deviation of 15, what is the minimum IQ
necessary for admission? - First: Find the z-score that corresponds to the given area using invNorm.
, z = invNorm(0.98, 0, 1)
x = μ + z * σ = 100 + 2.0537(15)
x = 130.8
x ≈ 131
mean of sampling distribution of x̄ - μx̄ = μ
The mean of the sampling distribution is denoted by μx̄ and is equal to the mean of the population μ.
standard deviation of sampling distribution of x̄ - σx̄ = σ/√n
The standard deviation of the sampling distribution (Standard Error) is denoted by σx̄ and equals the
standard deviation of the population divided by the square root of the sample size.
Sampling distribution of a skewed population - if a population is skewed, a larger sample size is necessary
for the sampling distribution of x̄ to be approximately normal.
The Central Limit Theorem - The theory that, as sample size increases, the distribution of sample means
of size n, randomly selected, approaches a normal distribution.
In practice, a sample size of n>30 is enough.
Example: Based on data from the U.S. Census, the mean age of college students in 2011 was μ = 25
years, with a standard deviation of σ = 9.5 years. A simple random sample of 125 students is drawn.
What is the probability that the sample mean age of the students is greater than 26 years? - Step 1:
Confirm the assumptions.
n=135>30
Thus, we may use the normal curve.
Step 2: Then, compute μx̄ and σx̄.
μx̄ = μ = 25
