Upcoming Schedule PSU Stat 2014

Upcoming SchedulePSU Stat 2014Monday Tuesday Wednesday Thursday Friday

Jan 6Sec 7.2

Jan 7 Jan 8Sec 7.3

Jan 9 Jan 10Sec 7.4

Jan 13

Chapter 7 in a nutshell

Jan 14 Jan 15

Chapter 7 test

Jan 16 Jan 17

Final Review

MLK Jr DayNo School

Jan 21

Monday schedule

Final Review

Jan 22

Final Per 1-3

Jan 23

Final Per 4-6

Jan 24

Final Per 7,8

JANUARY FINALS SCHEDULE:Here is the Finals Schedule for the end of the grading period in January.Tuesday, January 21, 2014 – Monday ScheduleWednesday, January 22, 2014                Period 1 Final                                          8:05-9:35                Period 2 Final                                          9:40-11:10                Lunch                                                      11:10-11:55                Period 3 Final                                          12:00-1:30                Proficiency Make Up Testing  1:35-3:05Thursday, January 23, 2014                Period 4 Final                                          8:05-9:35                Period 5 Final                                          9:40-11:10                Lunch                                                      11:10-11:55                Period 6 Final                                          12:00-1:30                Proficiency Make Up Testing  1:35-3:05Friday, January 24, 2014                Period 7 Final                                          8:05-9:35                Period 8 Final                                          9:40-11:10                Lunch                                                      11:10-11:55                Proficiency Make Up Testing  12:00-3:05Monday, January 27, 2014 – Teacher Planning Day

Confidence Interval of the Mean

Bluman, Chapter 6 3

y

μ2

2CI

α=1-CI

)2(

2

Z )

2(

2

Z

95% Confidence Interval of the Mean

Bluman, Chapter 7 4

Confidence Interval of the Mean for a Specific

)()(22 n

zxn

zx

CI z

90% 1.65

95% 1.96

98% 2.33

99% 2.58

Common confidence intervals, CI, and z scores associated with them.

7.2 Confidence Intervals for the Mean When Is Unknown

The value of , when it is not known, must be estimated by using s, the standard deviation of the sample.

When s is used, especially when the sample size is small (n<30), critical values greater than the values for are used in confidence intervals in order to keep the interval at a given level, such as the 95%.

These values are taken from the Student t distribution, most often called the t distribution.

Bluman, Chapter 7 6

2z

Characteristics of the t DistributionThe t distribution is similar to the standard normal distribution in these ways:

1. It is bell-shaped.

2. It is symmetric about the mean.

3. The mean, median, and mode are equal to 0 and are located at the center of the distribution.

4. The curve never touches the x axis.

Bluman, Chapter 7 7

Characteristics of the t DistributionThe t distribution differs from the standard normal distribution in the following ways:

1. The variance is greater than 1.

2. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size.

3. As the sample size increases, the t distribution approaches the standard normal distribution.

Bluman, Chapter 7 8

Degrees of Freedom The symbol d.f. will be used for degrees of

freedom. The degrees of freedom for a confidence

interval for the mean are found by subtracting 1 from the sample size. That is, d.f. = n - 1.

Note: For some statistical tests used later in this book, the degrees of freedom are not equal to n - 1.

Bluman, Chapter 7 9

The degrees of freedom are n - 1.

Formula for a Specific Confidence Interval for the Mean When IsUnknown and n < 30

Bluman, Chapter 7 10

2 2

s sX t X t

n n

Chapter 7Confidence Intervals and Sample Size

Section 7-2Example 7-5

Page #371

Bluman, Chapter 7 11

Find the tα/2 value for a 95% confidence interval when the sample size is 22.

Degrees of freedom are d.f. = 21.

Example 7-5: Using Table F

Bluman, Chapter 7 12

Chapter 7Confidence Intervals and Sample Size

Section 7-2Example 7-6

Page #372

Bluman, Chapter 7 13

Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.

Since is unknown and s must replace it, the t distribution (Table F) must be used for the confidence interval. Hence, with 9 degrees of freedom, tα/2 = 2.262.

Example 7-6: Sleeping Time

Bluman, Chapter 7 14

2 2

s sX t X t

n n

0.78 0.787.1 2.262 7.1 2.262

10 10

One can be 95% confident that the population mean is between 6.5 and 7.7 inches.

Example 7-6: Sleeping Time

Bluman, Chapter 7 15

0.78 0.787.1 2.262 7.1 2.262

10 10

7.1 0.56 7.1 0.56

6.5 7.7

Chapter 7Confidence Intervals and Sample Size

Section 7-2Example 7-7

Page #372

Bluman, Chapter 7 16

The data represent a sample of the number of home fires started by candles for the past several years. Find the 99% confidence interval for the mean number of home fires started by candles each year.

5460 5900 6090 6310 7160 8440 9930

Step 1: Find the mean and standard deviation. The mean is = 7041.4 and standard deviation s = 1610.3.

Step 2: Find tα/2 in Table F. The confidence level is 99%, and the degrees of freedom d.f. = 6

t .005 = 3.707.

Example 7-7: Home Fires by Candles

Bluman, Chapter 7 17

X

Example 7-7: Home Fires by Candles

Bluman, Chapter 7 18

Step 3: Substitute in the formula.

One can be 99% confident that the population mean number of home fires started by candles each year is between 4785.2 and 9297.6, based on a sample of home fires occurring over a period of 7 years.

1610.3 1610.37041.4 3.707 7041.4 3.707

7 7

2 2

s sX t X t

n n

7041.4 2256.2 7041.4 2256.2

4785.2 9297.6

Z or t; see page 373

Please read the paragraph on top of the page.

Bluman, Chapter 7 19

z or t, page 373

Bluman, Chapter 7 20

Is s known?

Use Za/2 values and s in the formula

Use ta/2 values and s in the formula

yes No

Homework

Sec 7.2 Page 374 #1-4 all and 5-19 every other odds

Optional: if you have a TI 83 or 84 calc see page 376

Bluman, Chapter 7 21