Warm Up 8/26/14 A study of college freshmen’s study habits found that the time (in hours) that...

Warm Up 8/26/14• A study of college freshmen’s study habits found that the time (in hours) that

college freshmen use to study each week follows a distribution with a mean of 7.2 hours and a standard deviation of 5.3 hours.

•  

a. Calculate the probability that a randomly chosen freshman studies more than 9 hours.

b. Find the probability that the average number of hours spent studying by an SRS of 55 students is greater than 9 hours. Show your work.

c. What are the mean and standard deviation for the average number of hours spent studying by an SRS of 55 freshmen?

How do I construct and interpret Confidence

Intervals using the margin of error?

Rate your confidenceRate your confidence0 - 1000 - 100

• Shooting a basketball at a wading pool, will make basket?

• Shooting the ball at a large trash can, will make basket?

• Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller?

The larger your confidence, the wider the interval.

Definition of a Confidence Interval• An interval that is computed from sample data and

provides a range of plausible values for a population parameter

• Is the success rate of the method used to construct the interval

• Formula: Mean of the sample + margin of error

Confidence Interval

Core Lesson

Construct: 65.25 ± 4.50 produces an interval from 60.75 to 69.75

Interpret: We are 95% confident the interval from $60.75 to $69.75 contains the actual mean amount of money teenagers spend on music per month.

Construct: 65.25 ± 4.50 produces an interval from 60.75 to 69.75

Interpret: We are 95% confident the interval from $60.75 to $69.75 contains the actual mean amount of money teenagers spend on music per month.

A sample of 150 teenagers finds the average amount of money spent on music per month is $65.25. With 95%

confidence, the margin of error is calculated to be $4.50. Construct and interpret the confidence interval.

Margin of errorMargin of error

• Shows how accurate we believe our estimate is

• The smaller the margin of error, the more precisemore precise our estimate of the true parameter

• Formula:

statistic theof

deviation standard

value

criticalm

Critical Values

• There are 4 typical levels of confidence: 99%, 98%, 95% and 90%.

Level of confidence Z – Critical Values

90% 1.645

95% 1.96

98% 2.33

99% 2.576

Confidence interval for a Confidence interval for a population mean:population mean:

n

zx

*

Estimate/mean of sample

Critical value

Standard deviation of the statistic

Margin of error

Steps for doing a confidence Steps for doing a confidence interval:interval:1) Assumptions –

• SRS from population• Sampling distribution is normal (or approximately

normal)• Given (normal)• Large sample size (approximately normal)• Graph data (approximately normal)

• is known

2) Calculate the interval

3) Write a statement about the interval in the context of the problem.

Statement (interpret): Statement (interpret): (memorize!!)(memorize!!)

We are ________% confident that the true mean context lies within the interval ______ and ______.

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

3899.3,0101.33

2.645.12.3

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

95% confidence interval?

4263.3,9737.23

2.96.12.3

99% confidence interval?

Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given) known

We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

4974.3,9026.23

2.576.22.3

What happens to the interval as the What happens to the interval as the confidence level increases?confidence level increases?

the interval gets wider as the confidence level increases

A random sample of 50 CHHS students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of CHHS students?

We are 95% confident that the true mean SAT score for CHS students is between 1220.9 and 1279.1

Suppose that we have this random sample of SAT scores:

950 1130 1260 1090 1310 1420 1190

What is a 95% confidence interval for the true mean SAT score? (Assume = 105)

We are 95% confident that the true mean SAT score for CHHS students is between 1115.1 and 1270.6.

In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.

For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.

Assumptions:

• Have an SRS of healthy males

• Systolic blood pressure is normally distributed (given).

• is unknown

We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.

)58.118,22.111(273.9

056.29.114

Warm Up 8/27/14

Research has shown that replacement times for TV sets have a mean of 8.2 years and a standard deviation of 1.1 years. He randomly selects a sample of 50 TV sets sold in the past and finds that the mean replacement time is 7.8 years.

•(a) Find the probability that 40 randomly selected TV sets will have mean replacement time of 7.8 years or less.

Why is the Central Limit Theorem important in statistics…

A sample of 300 high school students was asked how many text messages they sent on a given day. The mean was 38 with a standard deviation of 17.5.

(a) If we had 49 such samples, what would we expect the mean and standard deviation of the sampling distribution of means to be?

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.

(70.883, 74.497)

Ex. – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:

160 200 220 230 120 180 140

130 170 190 80 120 100 170

Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.

(126.16, 189.56)

Find a sample size:Find a sample size:

22

The heights of CHHS male students is normally distributed with = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?

n = 43