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Ch. 24 notes: Comparing 2 means: 2 sample t-interval and test
2-sample T-test
HYPOTHESES:
Ho: [pic]
Ha: [pic]
CONDITIONS:
Just double the 1- sample conditions…
1. 2 independent SRS
2. 10 % condition for both
[pic]
3. 2 Normal Populations OR [pic]
Conditions met ( t- distribution ( 2-sample t- Test
MECHANICS:
[pic]
P-Value: P(t test statistic l df = #)
To find the degrees of freedom –
On the calculator – round to the nearest whole number
OR
Use the smaller degrees of freedom of the two samples
CONCLUSION:
Same two sentences (includes the word “difference”)
- We reject/ fail to reject ….
- We have sufficient/insufficient evidence that the true difference in means is ….(use Ha)
Example: Resting pulse rates for a random sample of 26 smokers had a mean of 80 beats per minute (bpm) and a
standard deviation of 5 bpm. Among 32 randomly chosen nonsmokers, the mean and standard deviation were 74 and 6 bpm. Both sets of data were roughly symmetric and had no outliers. Is there evidence of a difference in mean pulse rate between smokers and non-smokers? How big?
[pic] [pic] [pic] [pic] [pic] [pic]
Ho: [pic]
Ha: [pic]
Conditions:
1. 2 independent SRS 1. Smokers and Non-smokers are independent
of one another
Both samples stated SRS
2. 10 % condition for both 2. There are more than 260 smokers and
more than 320 non-smokers so we are
sampling less than 10% of the populations
3. 2 Normal Populations OR [pic] 3. Both data sets were roughly symmetric
with no outliers so we can assume both are
nearly normal
Conditions met ( t- distribution ( 2-sample t- Test
[pic]
2* P(t > 4.154 l df = 56) = [pic]
We reject the Ho because the P-Value of [pic] < [pic].
We have sufficient evidence that the true average pulse rate (beats per minute) between smokers and non-smokers is different.
2-sample t-Interval
CONDITIONS:
Same as 2- sample t-Test
Conditions met ( t-distribution ( 2-sample t-Interval
MECHANICS:
statistic [pic](critical value)(standard deviation of statistic)
[pic]
INVT Difference of means
2-sample T-Interval (on calculator) Degrees of Freedom – from calculator
CONCLUSION:
We are ____% confident that the true difference between the mean of ______ & ________ is between ___a____ and __b____ units.
Example: Since we rejected our Ho, complete the interval for our example on smokers above.
Conditions met ( t- distribution ( 2- sample t- Interval df = 56
[pic]
We are 95% confident that the true difference between the average resting heart rate of smokers and non-smokers is between 3.106 bpm and 8.894 bpm.
Example 2: Here are the saturated fat content (in grams) for several pizzas sold by two national chains. Do the two pizza chains have significantly different mean saturated fat contents?
Brand D |17 |12 |10 |8 |8 |10 |10 |5 |16 |16 | | |8 |12 |15 |7 |11 |11 |13 |13 |11 |12 | |Brand PJ |6 |7 |11 |9 |4 |4 |7 |9 | | | | |11 |3 |4 |5 |8 |5 |5 | | | | |
Hypotheses:
[pic]
[pic]
[pic]
Conclusion:
We reject the Ho because the P-Value of .0000315 < [pic].
We have sufficient evidence that the average saturated fat content of brand D is not equal to brand PJ.
[pic]
Book Problems: p. 579 #2, 3, 11, 14, 27, 36
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