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Wize University Statistics Textbook > Continuous Probability Distributions

Standardizing

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Standard Normal Distribution (Z-Scores)

A continuous random variable ZZ follows a standard normal distribution if it's a special case of the normal distribution that is centered at 0 (i.e. μ=0\mu=0) and has a standard deviation of σ=1\sigma=1.
  • This is a VERY useful distribution, so we give it a special variable ZZ
  • We denote this by Z ∼ N(0, 1)Z\ \sim\ N\left(0,\ 1\right)
  • We use z-scores (zz) to denote the number of standard deviations a value is away from the mean
  • If z=0z=0, the value equals the mean
  • If z>0z>0, the value is to the right of the mean
  • If z<0z<0, the value is to the left of the mean
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Z-Tables

Standard normal distributions are very useful because we have a z-table (a.k.a. standard normal table) that gives us the area (i.e. probability) from the far left of the curve up to the z-value.

Examples
P(Z<1.32)=P(Z<1.32)=
0.9066

P(Z>1.32)=P\left(Z>1.32\right)=
1 - 0.9066 = 0.0934



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Examples
P(Z<−1.32)=P\left(Z<-1.32\right)=
0.0934

P(Z>−1.32)=P\left(Z>-1.32\right)=
1 - 0.0934 = 0.9066


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Wize Tip
Since the curve is symmetrical, some profs will only give you the z-table with positive z-scores.

In this case, draw a picture to help you visualize the area that corresponds to the desired probability.

Watch Out!
Since ZZ is a continuous random variable, its probability distribution function cannot be entirely represented by a table of discrete values. That's why the z-table only shows some of the values for the distribution. Sometimes you may have to approximate the probability.

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Standardizing a Normal Random Variable

Standard normal random variables are very useful since they allow us to use the z-table to solve probability problems.

What if you were asked to find the probability of a value associated with just a Normal random variable?

Standardization

Given a value associated with a normal random variable XX, we can find the z-score that corresponds to the standard normal random variable ZZ by standardizing it using this formula:
z=x−μσ\boxed{\displaystyle z=\frac{x-\mu}{\sigma}}
  • z\colorFour z is the z-score that follows the standard normal distribution ZZ
  • We can then use the z-table to solve probability problems!
  • x\colorFour x is the given value that follows the normal distribution XX
  • XX has a mean of μ\colorFour \mu
  • XX has a standard deviation of σ\colorFour \sigma
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Example
Suppose XX has a normal distribution with a mean of 500 and standard deviation of 250. What is the probability that XX is 830 or higher?

Using the standardization formula:
z=x−μσ=830−500250=1.32z=\dfrac{x-\mu}{\sigma}=\dfrac{830-500}{250}=1.32

This z-score of 1.32 tells us that value of 830 is 1.32 standard deviations above the mean, which is 500.

Using the z-table, the area (probability) to the left of 1.32 is 0.9066.

So,
P(Z>1.32)P(Z>1.32)
=1−0.9066=1-0.9066
=0.0934=0.0934

Therefore, the probability that XX has a value of 830 or higher is 0.0934.
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Practice: Standardizing

Contestants at a talent show were given scores between 0 and 100. The mean score is 65 with a standard deviation of 8. Assume the scores are normally distributed.

(a) Only the top 10% will compete in the finals. What is the minimum score to qualify?


P(Z<1.282)=0.9P\left(Z<1.282\right)=0.9

z=X−μσz=\frac{X-\mu}{\sigma}

1.282=X−6581.282=\frac{X-65}{8}

X=75.256X=75.256
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(b) What percent of the contestants received a score of 57 or lower?



z=X−μσ=57−658=−1\displaystyle{z=\frac{X-\mu}{\sigma}=\frac{57-65}{8}=-1}

z-table: P(Z<−1)=0.1587P\left(Z<-1\right)=0.1587

Empirical Rule: P(Z<−1)=1−0.682=0.16\displaystyle{P\left(Z<-1\right)=\frac{1-0.68}{2}=0.16}



(c) What percent of the contestants received a score between 57 and 89 percent?

See graph above.

z=X−μσ=89−658=3\displaystyle{z=\frac{X-\mu}{\sigma}=\frac{89-65}{8}=3}

z-table: P(Z>3)=1−0.9987=0.0013P\left(Z>3\right)=1-0.9987=0.0013

P(57<X<89)=P(−1<Z<3)=(1−0.0013)−0.1587=0.84P\left(57<X<89\right)=P\left(-1<Z<3\right)=\left(1-0.0013\right)-0.1587=0.84

Empirical Rule: P(Z<−1)=1−0.9972=0.0015\displaystyle{P\left(Z<-1\right)=\frac{1-0.997}{2}=0.0015}

P(57<X<89)=P(−1<Z<3)=1−0.0015−0.16=0.8385P\left(57<X<89\right)=P\left(-1<Z<3\right)=1-0.0015-0.16=0.8385

The class grade on the statistics final exam is normally distributed with a mean of 65 percent and a standard deviation of 8 percent.



Approximately 95% of all grades would lie between what two grades?


Skee Ball: In order to win a $1000 prize, Billy has to roll five balls and either score below 150 or over 250. Suppose the average player scores 210 with a standard deviation of 20. Assume Bill is an average player. What is the probability that he will not win a prize?

Enter answer with at least 2 decimal places (e.g. 0.88)
Extra Practice
Standardizing
When calculating students' marks out of 25 their z-scores are also calculated. The average score was 17 and the standard deviation was 1.4. Test scores are normally distributed. Eric tells John "my teacher said I scored higher than 60% of the class". What's the lowest mark Eric could have had?
Standardizing
IQ scores for 15 year olds are approximately normally distributed with mean 120 and standard deviation 16. Mandy's IQ score is 115.
Standardizing
George is a Shiba Inu breeder. The average weight of an adult is 20 lbs with a standard deviation of 5.5lbs. A Shiba Inu is considered overweight if it weighs more than 28 lbs, and underweight if it weighs less than 16 lbs. What proportion of Shiba Inus are normal weight?
Standardizing
The waiting time for customer at the Beg Steakhouse and Bar is approximately normal with mean 19 minutes. 3% of customers have to wait longer than 22 minutes. Find the standard deviation.
Standardizing
Find the third quartile of a normal distribution with mean 54 and standard deviation 12.
Standardizing
When calculating students' marks out of 25 their z-scores are also calculated. The average score was 17 and the standard deviation was 1.4. Test scores are normally distributed. Eric tells John "my teacher said I scored higher than 60% of the class". What's the lowest mark Eric could have had?
Standardizing
The average time for Papa Dough to deliver a pizza in downtown is 45 minutes with a standard deviation of 8 minutes. The restaurant policy states that the pizza is free if it is not delivered within one hour.
Assuming the delivery time is normally distributed, what percentage of pizza are free?
Standardizing
IQ scores for 15 year olds are approximately normally distributed with mean 120 and standard deviation 16. Mandy's IQ score is 115.
What proportion of 15 year olds have an IQ score below Mandy's IQ?
Standardizing
In Japan, your entrance exam score determines what university you get into; the higher the entrance exam score, the more prestigious the school you get to go to. Suppose each year, the top 5% goes to BYC University, the best school in Japan. The bottom 30% does not enter university. This year, the mean entrance exam score is 450 with a standard deviation of 65. What is the minimum entrance grade to enter BYC University?
Standardizing
Max and his friend, Caroline, play 10-pin bowling every Friday. Max's average score is 260 with a standard deviation of 8. Caroline's average score is 241 with a standard deviation of 14. Assume that the scores for both players are normally distributed. What proportion of the times will Max score above Caroline's mean this Friday?
Standardizing
The number of births at a hospital varies from month to month with average of 60 and variance 25. Assuming this distribution is approximately normal, what proportion of the days do we expect more than 69 births?
Standardizing
The number of calories in a muffin at a fast food restaurant is approximately normal with a mean of 800 and standard deviation 15. What percent of muffins between 800 and 834 calories?
Standardizing
The average time for Papa Dough to deliver a pizza in downtown is 45 minutes with a standard deviation of 8 minutes. The restaurant policy states that the pizza is free if it is not delivered within one hour.
Standardized Score
Suppose that the measure of an "awesomeness score" for kids at a daycare follow a normal distribution model. If 5-year-old Andrew has a standardized score (z-score) of +1.2, which of the following statements is TRUE?
Standardizing
The waiting time for customer at the Beg Steakhouse and Bar is approximately normal with mean 19 minutes. 3% of customers have to wait longer than 22 minutes. Find the standard deviation.
Standardizing
The average time for Papa Dough to deliver a pizza in downtown is 45 minutes with a standard deviation of 8 minutes. The restaurant policy states that the pizza is free if it is not delivered within one hour.
Standardizing
IQ scores for 15 year olds are approximately normally distributed with mean 120 and standard deviation 16. Mandy's IQ score is 115.
Standardizing
In Japan, your entrance exam score determines what university you get into; the higher the entrance exam score, the more prestigious the school you get to go to. Suppose each year, the top 5% goes to BYC University, the best school in Japan. The bottom 30% does not enter university. This year, the mean entrance exam score is 450 with a standard deviation of 65. What is the minimum entrance grade to enter BYC University?
Proportions: Standardizing
Max and his friend, Caroline, play 10-pin bowling every Friday. Max's average score is 260 with a standard deviation of 8. Caroline's average score is 241 with a standard deviation of 14. Assume that the scores for both players are normally distributed. What proportion of the times will Max score above Caroline's mean this Friday?
Proportions: Standardizing
The number of births at a hospital varies from month to month with average of 60 and variance 25. Assuming this distribution is approximately normal, what proportion of the days do we expect more than 69 births?
Standardizing
Suppose that the measure of an "awesomeness score" for kids at a daycare follow a normal distribution model. If 5-year-old Andrew has a standardized score (z-score) of +1.2, which of the following statements is TRUE?
Working Backwards to Find Z-Score
ZZ is a standard normal random variable.
Standardizing
Darlene is in Professor Kitt’s class. She got 70 on her midterm, which had a mean score of 60 and a standard deviation of 20. Her boyfriend, Jacob – a history major – got 78 on his HIST 255 midterm and brags that he is smarter. The HIST 255 midterm has a mean of 65 with a standard deviation of 26. Who is in fact smarter relative to their classes?
Standardizing
Professor Kitts gave his MATH 184 class a hard midterm. If 30.85% of the class got less than 50 and 11.51% of the class got higher than 84, find the mean and standard deviation.
Standardizing
George is a Shiba Inu breeder. The average weight of an adult is 20 lbs with a standard deviation of 5.5lbs. A Shiba Inu is considered overweight if it weighs more than 28 lbs, and underweight if it weighs less than 16 lbs. What proportion of Shiba Inus are normal weight?
Standardizing
Find the third quartile of a normal distribution with mean 54 and standard deviation 12.
Standardizing
The number of calories in a muffin at a fast food restaurant is approximately normal with a mean of 800 and standard deviation 15. What percent of muffins have between 800 and 834 calories?
Standardizing
The average time for Papa Dough to deliver a pizza in downtown is 45 minutes with a standard deviation of 8 minutes. The restaurant policy states that the pizza is free if it is not delivered within one hour.
If Papa Dough wants to only make 1% of pizzas free, then what should be the approximate cutoff delivery time now?
Properties
IQ scores for 15 year olds are approximately normally distributed with mean 120 and standard deviation 16. Mandy's IQ score is 115.
According to the empirical rule, what proportion of 15 year olds have an IQ score between 88 and 152?
Quartiles Normal Distributions
Answer the following 2 questions about normal distributions.
Standardizing
Sony is interested in the number of songs on a person’s iPhone. The distribution of songs is normal with a mean of 300 and a standard deviation of 60.
(a) What’s the probability that a random iPhone will have more than 375 songs?
(b) If Sony randomly samples 12 iPhones, what’s the probability that the sample mean will be more than 375 songs?
Standardizing
The Pogs Fan Club only has 150 members. The number of pogs a member owns is normally distributed with a mean of 265 and standard deviation of 35. What is the probability of selecting a random sample of 10 members with a sample mean of 280 or more?
Standardizing
Kick the Butt is interested in the number of cigarettes a smoker smokes a day. The distribution of daily number of cigarettes is normal with a mean of 15 and a standard deviation of 8.
1. What’s the probability that a random smoker smokes more than 20 cigarettes a day?
2. If they randomly sample 10 smokers, what’s the probability that the sample mean will be more than 13 cigarettes a day?
Standardizing
Refer to Question 15.
What is the standard deviation?
Standardizing
The waiting time for customer at the Beg Steakhouse and Bar is approximately normal with mean 19 minutes. 3% of customers have to wait longer than 22 minutes. Find the standard deviation.
Standardizing
Find the third quartile of a normal distribution with mean 54 and standard deviation 12.
Standardizing
George is a Shiba Inu breeder. The average weight of an adult is 20 lbs with a standard deviation of 5.5lbs. A Shiba Inu is considered overweight if it weighs more than 28 lbs, and underweight if it weighs less than 16 lbs. What proportion of Shiba Inus are normal weight?
Standardizing
The exam grades in Hugh’s statistics class are normally distributed. If his z-score is 0.00, which of the following must be true?
Standardizing
Bayleaf Airport Lounge has an average of 85 patrons a day with a standard deviation of 15 patrons.
(a) On Saturdays, the number of patrons always exceeds 100 patrons. Given that, what is the probability that there will be at least 118 patrons this upcoming Saturday?
(b) The lounge can only accommodate 130 guests on any given day, from which 10 lounge passes are put aside for VIP guests like celebrities and politicians. What is the probability that there will not be enough lounge passes?
Standardizing
The class grade on the statistics final exam is normally distributed with a mean of 65 percent and a standard deviation of 8 percent.
Standardizing: Sample Distribution for a Mean
We do not know if population distribution is skewed or not but we know that the population mean is 50 and the population standard deviation is 16. What is the probability that the sample mean will be larger than 54 for a sample size of 36?
Standardizing
Ash and Jin both took STAT 100 in different sections. Grades in both sections are normally distributed. The class average in Ash's section is 68% with a standard deviation of 10%. Ash got 76% in the course. Jin's grade is one standard deviation above his section's average grade. The person with who is ranked lower in their respective sections buys dinner. Which of the following is true?
Inference Statistics
The random variable XXhas a normal distribution. If μ=100\mu=100and σ=17\sigma=17, then P(X>100+d)=0.1230P\left(X>100+d\right)=0.1230. Find the value of dd, rounding to the nearest integer.
Empirical Rule
Contestants at a talent show were given scores between 0 and 100. The mean score is 65 with a standard deviation of 8. Assume the scores are normally distributed.
For parts (b) and (c) provide all your answer in decimal format with at least 2 decimal places. (For example, 32.56% should be entered as 0.33, 0.326, or 0.3256.)
Standardizing
Skee Ball: In order to win a $1000 prize, Billy has to roll five balls and either score below 150 or over 250. Suppose the average player scores 210 with a standard deviation of 20. Assume Bill is an average player. What is the probability that he will not win a prize?
Enter answer with at least 2 decimal places (e.g. 0.88)
Standardizing
Max and her frenemy, Caroline, play 10-pin bowling every Friday. Max’s average score is 260 with a standard deviation of 8. Caroline’s average score is 235 with a standard deviation of 14.
Enter your answers with at least 2 decimal places (e.g. 0.89)
Standardizing
Ben is hosting a house party at his new loft. The number of beers a guest consumes is normally distributed with a mean of 3.3 and a standard deviation of 1.6.
What’s the probability that a random party guest consumes more than 4 beers?
Standardizing: Normal Distribution
The wine ratings on a website is normally distributed with a mean of 3.70 and a standard deviation of 0.65. Seventy-five percent of the wine ratings are greater than:
Statistical sampling
Entering Crowne Academy is highly competitive, and those who wish to enroll must take a standard test called CUEE (Crowe University Entrance Exam). Last year, the average grade of students who took the CUEE was 75% with a standard deviation of 12%.
If we take a sample of 10 students who took the CUEE, what is the probability that their mean grade is above 80%?
Statistical Properties: Practice
True or false? Enter "T" for true; "F" for false (without quotations).
Standardizing
Suppose the time spent at the gym is normally distributed with a mean of 50 minutes and a standard deviation of 15 minutes.
What percent of gym members spends more than an hour at the gym?
Standardizing: Sampling Distribution
Half of the population can swim. There is a 19.77% probability that that more than p^\hat{p}% of my sample of 50 can swim. Find xx.
Normal distribution and standardizing
The random variable X has a normal distribution. If μ=20μ=20 and σ=4σ=4 find the value of aa such that P(16<X<a)=0.7357P(16<X<a)=0.7357. [Round aa to the nearest whole number.]
Normal Distribution and Standardizing
If X = 200, we get z = 2.00; and
if X = 115, we get z = –1.40
Solve for μ\mu and σ\sigma.
Standardizing: Normal Distribution
The exam grades in Hugh’s statistics class are normally distributed. If his z-score is 0.00, which of the following must be true?
Normal distribution: Standardizing
The average tips servers receive is normally distributed with a mean of 13.5% and a standard deviation of 4.5%. What is the probability of Trevor receiving more than 15% on a given bill?
Normal distribution
A random skills test was administered to a large group of students at a university. Suppose that the time the students took to complete the test is approximately normal with a mean of 40 minutes and a standard deviation of 10 minutes. If the top 20% of the students who completed the skills test the fastest get a prize, how fast must a candidate complete the test to get a prize?
Finding Probability From Z-Score
ZZ is a standard normal random variable.
Normal Distribution: Standardizing
When calculating students' marks out of 25, their z-scores are also calculated. The average score was 17 and the standard deviation was 1.4. Test scores are normally distributed. Eric tells John "my teacher said I scored higher than 60% of the class". What's the lowest mark Eric could have had?