• Red line - true density
• Blue dashed line - normal model with mean and SD from observed heights

The normal distribution is the most common distribution in statistics. It has a mean \(\mu\) and a standard deviation \(\sigma\), which describe it entirely.

If we assume a random variable \(X\) is normally distributed (mean \(\mu\), SD \(\sigma\)), then

• \(X\sim N(\mu,\sigma)\)
• Distribution of \(X\) is:
  • Unimodal
  • Symmetric
  • “Bell-shaped”

The Z-score of an observation is the number of standard deviations it falls above or below the mean. The Z-score for an observation \(x\) that follows a distribution with mean \(\mu\) and standard deviation \(\sigma\) is computed as:

\[Z=\frac{x-\mu}{\sigma}\]

The normal distribution with \(\mu=0\) and \(\sigma=1\) is called the standard normal distribution.

• Facilitate comparisons
  • Percentiles for standardized testing, SAT vs ACT
  • Percentiles for height and weight for toddlers in different populations
• Z-scores have a special meaning
  • number of standard deviations above or below the mean

If \(X \sim N(\mu, \sigma)\), then \(X\) is a continuous random variable (recall Section 3.5).

What does probability mean for a continuous random variable?

If \(X \sim N(\mu, \sigma)\), then \(X\) is a continuous random variable (recall Section 3.5).

What does probability mean for a continuous random variable?

• Defined as area under the curve (e.g. \(P(69\leq X \leq 73)\)):

How do we find the area under the curve (probability) if \(X \sim N(\mu,\sigma)\)?

1. Calculus (not covered in this course)
2. R: pnorm() function
3. Normal table

For (2) and (3), we assume our process follows a normal distribution - this is a MODEL - it is NOT EXACT.

pnorm() gives us the probability of an observation below a certain value, given an appropriate mean and SD

## P(X < 69):
pnorm(69, mean=mean(cdc_m$height), sd=sd(cdc_m$height))

## [1] 0.338728

## P(X < 72):
pnorm(72, mean=mean(cdc_m$height), sd=sd(cdc_m$height))

## [1] 0.7193796

Z table

According to the CDC data base, the mean height of US men is 70.25 inches and the SD is 3.01 inches. If we model height (random variable \(X\)) as normal (with the previously stated mean and variance), what is the probability a male is between 69 and 72 inches tall?

According to the CDC data base, the mean height of US men is 70.25 inches and the SD is 3.01 inches. If we model height (random variable \(X\)) as normal (with the previously stated mean and variance), what is the probability a male is between 69 and 72 inches tall?

Step 1: Draw a picture to identify what you want

Step 2: Identify what you information you can get

## P(X < 69)
pnorm(q=69, 
      mean=mean(cdc_m$height), 
      sd=sd(cdc_m$height))

## [1] 0.338728

## P(X < 72)
pnorm(q=72, 
      mean=mean(cdc_m$height), 
      sd=sd(cdc_m$height))

## [1] 0.7193796

Step 3: Connect what we have to what we want

Both pnorm() and the Z table give lower tail probabilities. To get what we want:

\[P(69 \leq X \leq 72)=P(X\leq 72)-P(X\leq 69)=0.7914-0.3387=0.4527\]

At Heinz ketchup factory the amounts which go into bottles of ketchup are supposed to be normally distributed with mean 36 oz. and standard deviation 0.11 oz. Once every 30 minutes a bottle is selected from the production line, and its contents are noted precisely. If the amount of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the bottle fails the quality control inspection. What percent of bottles have less than 35.8 ounces of ketchup?

At Heinz ketchup factory, the amounts which go into bottles of ketchup are supposed to be normally distributed with mean 36 oz. and standard deviation 0.11 oz. Once every 30 minutes a bottle is selected from the production line, and its contents are noted precisely. If the amount of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the bottle fails the quality control inspection. What percent of bottles have less than 35.8 ounces of ketchup?

At Heinz ketchup factory, the amounts which go into bottles of ketchup are supposed to be normally distributed with mean 36 oz. and standard deviation 0.11 oz. Once every 30 minutes a bottle is selected from the production line, and its contents are noted precisely. If the amount of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the bottle fails the quality control inspection. What percent of bottles have less than 35.8 ounces of ketchup?

\(P(X<35.8)=P(X\leq 35.8)=\) 0.0345182

What percent of bottles pass quality control inspection?

What percent of bottles pass quality control inspection?

What percent of bottles pass quality control inspection?

## P(X < 36.2)-P(X < 35.8)
pnorm(q=36.2, mean=36, sd=0.11)-pnorm(q=35.8, mean=36, sd=0.11)

## [1] 0.9309637

Body temperatures of healthy humans are distributed nearly normally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the lowest 3% of human body temperatures?

• Use R: qnorm() function (what does this return?)

qnorm(0.03)

## [1] -1.880794

• Use Z table (work backwards)

Body temperatures of healthy humans are distributed nearly normally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the lowest 3% of human body temperatures on the original scale?

• Use R: qnorm() function (what does this return?)

co <- qnorm(0.03)
co

## [1] -1.880794

Body temperatures of healthy humans are distributed nearly normally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the lowest 3% of human body temperatures on the original scale?

• Use R: qnorm() function (what does this return?)

co <- qnorm(0.03)
co

## [1] -1.880794

\(Z=\frac{x-\mu}{\sigma}=\frac{x-98.2}{0.73}=-1.88\)

\(-1.88\times 0.73+98.2=96.8\)

Body temperatures of healthy humans are distributed nearly normally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the highest 10% of human body temperatures (on the original scale)?

Body temperatures of healthy humans are distributed nearly nor- mally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the highest 10% of human body temperatures (on the original scale)?

co <- qnorm(0.90)
co

## [1] 1.281552

Body temperatures of healthy humans are distributed nearly nor- mally with mean 98.2 F and standard deviation 0.73 F. What is the cutoff for the highest 10% of human body temperatures (on the original scale)?

co <- qnorm(0.90)
co

## [1] 1.281552

\(Z=\frac{x-\mu}{\sigma}=\frac{x-98.2}{0.73}=1.28\)

\(1.28\times 0.73+98.2=99.1\)

Rule of thumb for the probability of falling within 1, 2, and 3 standard deviations of the mean in the normal distribution.

SAT scores are distributed nearly normally with mean 1500 and standard deviation 300.

SAT scores are distributed nearly normally with mean 1500 and standard deviation 300.

Which of the following is false?

1. Majority of Z scores in a right skewed distribution are negative.
2. In skewed distributions the Z score of the mean might be different than 0.
3. For a normal distribution, IQR is less than 2 x SD.
4. Z scores are helpful for determining how unusual a data point is compared to the rest of the data in the distribution.

Which of the following is false?

1. Majority of Z scores in a right skewed distribution are negative.
2. In skewed distributions the Z score of the mean might be different than 0.
3. For a normal distribution, IQR is less than 2 x SD.
4. Z scores are helpful for determining how unusual a data point is compared to the rest of the data in the distribution.

• Diez et al. (2019) OpenIntro Statistics, Fourth Edition