# Sampling Distribution Formed by Sampling from More Than One Normal Distribution

**Sampling Distribution **

**Formed by Sampling from More Than One Distribution**

GOAL: Determine the mean and standard deviation of the sampling distributions as described below. Sketch and label the sampling distribution. Then determine the probability of event described.

**CENTRAL LIMIT THEOREM**

**Sampling Distribution of One Sample Proportion Sampling Distribution of One Sample Mean**

SPECIAL CASES

**Sampling Distribution of Differences between two independent sample means**

**Sampling Distribution of Differences between two independent sample proportions**

OTHERWISE

Rely on Linear Combinations Rules (and Linear Transformation Rules)

NOTE: To work the following problems, refer to Sampling Distribution with CLT paper.

- A student drives over a quarter several times with his car tires. He wants to determine if the coin is still fair. He flips the coin 100 times. How likely is it that 42% of the time or fewer he would have gotten tails? Remember: p (also known as ) is NOT the same as .

- The same flipped student runs over another quarter several times with his car tires. He wants to determine if there is a difference between the proportions of tails. He flips each coin 100 times and finds the first quarter to have landed tails 44% of the time and the second quarter to have landed tails 53% of the time. How likely would this difference or more extreme be?

- Each year AP Statistics students at OHS attempt to guess Mrs. Gann’s age. Assuming through the years, individual students’ kind guesses model a normal distribution with an estimate of 35 years of age and standard deviation of 2 years. If the prior information is correct (assume so, or fail), what is the likelihood that this year’s class of 92 students will estimate her age at 37 years or greater? Assuming she never ages (as her secretary was instructed to type).

- Since Mrs. Gann is now a grandmother and Ms. Uzzell is not, we will assume the actual difference in their ages is zero. Over the past several years, the standard deviation of estimated age for Ms. Uzzell is 3 years. See #3 for Mrs. Gann info. Using the same number of students to guess at their ages, what is the likelihood the estimated difference between Gann’s minus Uzzell’s estimated age is 2 years or more?

- A company manufactures small stereo systems. At the end of the production line, the stereos are packaged and prepared for shipping independently. Stage 1 of this process is called “packing.” Workers must collect all the system components, put each in plastic bags, and then place everything inside a protective Styrofoam form. The packed form then moves on to stage 2, called “boxing.” There workers place the form and a packet of instructions in a cardboard box, close it, then seal and label the box for shipping. The company says that times required by the packing stage can be described by a Normal model with a mean of 9 minutes and standard deviation of 1.5 minutes. The times for the boxing stage can also be modeled as Normal, with a mean of 6 minutes and standard deviation of 1 minute. What is the likelihood that the total time to package and box exceeds 18 minutes? Note: sample size, n, is 1.

- In a 4x100 medley relay event, four swimmers swim 100 yards, each using a different stroke. OHS team preparing for the state championship looks at the times of their swimmers have posted and creates a model based on the following assumptions:
- The swimmers’ performances are independent.
- Each swimmer’s times follow a Normal model.
- The means and standard deviations of the times (in seconds) are as shown:

Swimmer / Mean / Std. Dev.

1. Backstroke / 50.72 / 0.24

2. Breaststroke / 55.51 / 0.22

3. Butterfly / 49.43 / 0.25

4. Freestyle / 44.91 / 0.21

The team’s best time so far this season was 3:19.48. (That’s 199.48 seconds.) Do you think the team is likely to swim faster than this at the state championship? Note: sample size is 1 for each swimmer style.