Upcoming Schedule PSU Stat 2014 Monday Tuesday Wednesday Thursday Friday Jan 6 Sec 7.2 Jan 7 Jan 8 Sec 7.3 Jan 9 Jan 10 Sec 7.4 Jan 13 Chapter 7 in a nutshell Jan 14 Jan 15 Chapter 7 test Jan 16 Jan 17 Final Review MLK Jr Day No School Jan 21 Monday schedule Jan 22 Final Per 1-3 Jan 23 Final Per 4-6 Jan 24 Final Per 7,8
Upcoming Schedule PSU Stat 2014. JANUARY FINALS SCHEDULE: Here is the Finals Schedule for the end of the grading period in January. Tuesday, January 21, 2014 – Monday Schedule Wednesday, January 22, 2014 Period 1 Final 8:05-9:35 - PowerPoint PPT Presentation
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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