WARM – UP 1. What is the Critical Value t* for a distribution of 26

observation with probability 0.10 to the Right?

2. What is the Critical Value t* for a 98%Confidence Interval of 4 observations?

3. What is the p-value for a one-sample t-Test with H0: μ = 3 vs. Ha: μ > 3 , n = 27 and t = 2.98

4. What is the p-value for a one-sample t-Test with H0: μ = 4 vs. Ha: μ ≠ 4 , n = 7 and t = -1.75

1.3161.316

4.5414.541

0.00310.0031

0.13070.1307

WARM – UP

In a study of heart surgery, one issue was the effects of drugs called beta blockers on the pulse rate of patients during surgery. Subjects were randomly divided into two groups of 30 patients each. One group received a beta blocker, the other a placebo. Pulse rates were recorded during the operations. The treatment group had a mean of 65.2 beats/min. (s = 7.8), while the control group was 70.3 beats/min. (s = 8.3).

Give a 95% confidence Interval for the mean difference.

Treatment:Treatment: 65.2 65.2 7.87.8Control:Control: 70.3 70.3 8.38.3

x s

μμ11 = The true mean number of beats/min for the treatment group. = The true mean number of beats/min for the treatment group.

μμ22 = The true mean number of beats/min for the control group. = The true mean number of beats/min for the control group.

x x ts

n

s

n1 212

1

22

2

b g 2 27.8 8.3

65.2 70.330 30

2.045

We can be 95% confident that the true mean difference in the number We can be 95% confident that the true mean difference in the number of beats/min is between -9.263 and -0.937 (of beats/min is between -9.263 and -0.937 (ββ Blockers – Placebo) . Blockers – Placebo) .

1.1. SRS – Stated SRS – Stated 2. Independent2. Independent3. Approximately Normal Distribution – YES because both 3. Approximately Normal Distribution – YES because both

sample sizes are ≥ 30 [Central Limit Theorem]sample sizes are ≥ 30 [Central Limit Theorem]

TWO Sample

TWO Sample

t – Conf. Int.

t – Conf. Int.

Treatment:Treatment: 65.2 65.2 7.87.8Control:Control: 70.3 70.3 8.38.3

x s

HH00: : μμ11 = = μμ22

HHaa: : μμ11 < , >, < , >, oror ≠ ≠ μμ22

tx x

sn

sn

1 2

12

1

22

2

The Hypothesis:

The Mechanics:

The t Test Statistic =

Ha: Ha: μμ11 < < μμ2 2 tcdf tcdf(-E99, (-E99, tt, , dfdf))

Ha: Ha: μμ11 > > μμ2 2 tcdftcdf((tt, E99, , E99, dfdf))

Ha: Ha: μμ11 ≠ ≠ μμ2 2 22··tcdftcdf(|(|t|t|, E99, , E99, dfdf))

CH. 24 - The Two Sample t – Test

The Conservative

Degree of freedom: df = ndf = n(smallest)(smallest) – 1 1

The P-Value =

In a study of heart surgery, one issue was the effects of drugs called beta blockers on the pulse rate of patients during surgery. Subjects were randomly divided into two groups of 30 patients each. One group received a beta blocker the other a placebo. Pulse rates were recorded during the operations. The treatment group had a mean of 65.2 beats/min. (s = 7.8), while the control group was 70.3 beats/min. (s = 8.3). Is there evidence that beta blockers reduce pulse rate?

HH00: : μμ11 = = μμ22

HHaa: : μμ11 < < μμ22

tx x

sn

sn

1 2

12

1

22

2

2 2

65.2 70.3

7.8 8.330 30

t

2.453t

2.453 99, 2.453,29 0.0086P Value P T tcdf E

Since the P-Value is less than Since the P-Value is less than αα = 0.01 the data IS significant . There = 0.01 the data IS significant . There is strong evidence to REJECT His strong evidence to REJECT H00 . Beta blockers do reduce pulse. . Beta blockers do reduce pulse.

TWO Sample

TWO Sample

t – Testt – Test

μμ11 = The true mean number of beats/min for the treatment group. = The true mean number of beats/min for the treatment group.

μμ22 = The true mean number of beats/min for the control group. = The true mean number of beats/min for the control group.

1.1. SRS – Stated SRS – Stated 2. Independent2. Independent3. Approximately Normal Distribution – YES because both 3. Approximately Normal Distribution – YES because both

sample sizes are above 30 [Central Limit Theorem]sample sizes are above 30 [Central Limit Theorem]

Does Buffalo, New York, receive significantly more snow than Does Buffalo, New York, receive significantly more snow than Chicago, Illinois? To test this question, researchers randomly Chicago, Illinois? To test this question, researchers randomly collected data of yearly snowfall amounts in each city. What collected data of yearly snowfall amounts in each city. What conclusions can be made from this data?conclusions can be made from this data?

Buffalo:Buffalo: 78” 130” 140” 120” 108” 120” 156” 101” 78” 130” 140” 120” 108” 120” 156” 101”Chicago:Chicago: 84” 96” 114” 81” 98” 103”84” 96” 114” 81” 98” 103”

μμ11 = The true mean amount of annual Snow fall in Buffalo, NY. = The true mean amount of annual Snow fall in Buffalo, NY.

μμ22 = The true mean amount of annual Snow fall in Chicago, IL. = The true mean amount of annual Snow fall in Chicago, IL.

HH00: : μμ11 = = μμ22

HHaa: : μμ11 > > μμ22

tx x

sn

sn

1 2

12

1

22

2

t

119125 96

24 088

12 216

2 2

.

. . t 2 3438.

2.3438 2.3438, 99,5 0.0196P Value P T tcdf E

Since the P-Value is less than Since the P-Value is less than αα = 0.05 the data IS significant . There = 0.05 the data IS significant . There is strong evidence to REJECT His strong evidence to REJECT H00 . Buffalo’s yearly snowfall average . Buffalo’s yearly snowfall average

is greater than that of Chicago, ILis greater than that of Chicago, IL1.1. SRS – Both Stated SRS – Both Stated 2. Independent2. Independent3. Approximately Normal Distribution – 3. Approximately Normal Distribution – Graph BOTHGraph BOTH! !

TWO Sample

TWO Sample

t – Testt – Test

Is there evidence to suggest that Mr. Belin’s Period 3 and Is there evidence to suggest that Mr. Belin’s Period 3 and Period 4 AP Statistics Classes performed differently on Period 4 AP Statistics Classes performed differently on Quiz #1? To test this an SRS of students were selected Quiz #1? To test this an SRS of students were selected from each class. from each class.

Period 3:Period 3: 86 100 98 72 100 86 98 100 86 100 98 72 100 86 98 100Period 4:Period 4: 68 94 98 82 80 83 88 100 76 9268 94 98 82 80 83 88 100 76 92

μμii = The true mean Quiz score of Mr. Belin AP Statistic class (Period 3 = The true mean Quiz score of Mr. Belin AP Statistic class (Period 3

and Period 4).and Period 4).

HH00: : μμ33 = = μμ44

HHaa: : μμ33 ≠ ≠ μμ44

tx x

sn

sn

1 2

12

1

22

2

2 2

92.5 86.1

10.24 10.168 10

t

1.3222t

2 1.3222 2 1.3222, 99,7 0.2058P Value P T tcdf E

Since the P-Value is NOT less than Since the P-Value is NOT less than αα = 0.05 the data IS Not = 0.05 the data IS Not significant. There is NO evidence to REJECT Hsignificant. There is NO evidence to REJECT H00 . Period 3 and Period . Period 3 and Period

4 perform similarly on quiz #1.4 perform similarly on quiz #1.1.1. SRS – Stated SRS – Stated 2. Independent2. Independent3. Approximately Normal Distribution – 3. Approximately Normal Distribution – Graph BOTHGraph BOTH! !

TWO Sample

TWO Sample

t – Testt – Test

Page 566:

#1 The Core Plus Mathematics Project (CPMP) in an innovative approach to teaching Math to kids by engaging them in group investigation and math modeling. 320 Students from the CPMP program and 273 from traditional curriculum were given an algebra test, the scores were recorded, and a 95% confidence interval was computed for the mean difference between the results: [ 5.573, 11.427 ]

a.) Margin of Error?

b.) Would a 98% be larger or smaller?

c.) Interpret

d.) Is the CPMP program better?

2 21 2

1 21 2

s sx x t

n n

HOMEWORK: Page 566: 3, 4, 9, 10, 12, 16

#3 The Core Plus Mathematics Project (CPMP) also gave students tests and did NOT allow calculators. The results are indicated below. Are the mean scores of the 2 groups significantly different?

a.) Hypothesis

b.) Assumptions SRS, Appr. Norm. via the CLT (large n)

c.&d.) Conclusion.

CPMP:CPMP: 29.0 29.0 18.8 31218.8 312Traditional:Traditional: 38.4 38.4 16.2 26516.2 265

x s n

μμ11 = The true mean score for 1=CPMP, 2=Trad. = The true mean score for 1=CPMP, 2=Trad.

HH00: : μμ11 = = μμ22

HHaa: : μμ11 ≠ ≠ μμ22