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Lab 7: The Normal 2-Sample Location Problem

1.
Consider the following data set consisting of atomic weights of carbon found by 2 different methods:

Method A Method B
12.0072 11.9853
12.0064 11.9949
12.0054 11.9985
12.0016 12.0061
12.0077  

Hypotheses: Let $\mu_{A}$ denote the population mean for method A, and $\mu_{B}$ the population mean for method B. We want to test $H_{0}:\mu_{A}=\mu_{B}$ vs. $H_{1}:\mu_{A} \ne \mu_{B}$, and we note that this is equivalent to testing: $H_{0}:\Delta = 0$ vs. $H_{1}:\Delta \ne 0$ where $\Delta = \mu_{A}-\mu_{B}$.

2.
Now, turn ``By Groups'' off. Perform Welch's approximate t-test using a 2-sided significance level of $\alpha=0.05$. To do this, select ``t-test'' under the ``Stats'' option of the ``Stats'' menu. Select the ``independent'' option in the lower right corner of the window. Then select WEIGHT from the variables box and METHOD$ from the group box. SYSTAT will then perform two t-tests. The pooled variance t-test is Students', whereas the separate variance t-test is Welch's.

3.
Calculate the 2-sided 95% confidence interval for $\Delta$. To determine the critical value, t0.975,df, use the TIF($\alpha$,df) function under the ``Math'' menu. In order to do this, let TVAL=TIF($\alpha$,df). Note, you are looking for a 2-sided c.i., so choose the $\alpha$ appropriately. Also, when looking at the formula on the first page, be sure to note that $t_{(1-\alpha/2),df}$ is the critical value, not the value of Welch's test statistic that was asked for in part a of question 2.

4.
What can be inferred about the two methods for determining the atomic weight of carbon?

5.
The second data set we will consider today is data on the birth weights of smoking and non-smoking mothers. It is believed that babies of nonsmoking mothers tend to have higher birth weights than babies of smoking mothers.



DATA: The data are from a study of the birth weights in pounds of the children of 40 mothers who smoked at least one pack a day during pregnancy and 39 mothers who did not smoke at all.

10cChildren of Nonsmoking Mothers                  
8.3 7.9 9.6 7.1 6.8 10.2 7.3 8.8 8.0 9.5
5.9 10.1 8.2 8.7 9.6 12.3 8.1 7.3 7.8 6.6
9.1 7.4 6.8 7.5 8.2 6.6 7.9 8.4 8.9 10.4
9.0 7.5 8.2 8.7 7.0 10.8 9.9 8.8 12.3  

10cChildren of Smoking Mothers                  
8.1 6.5 7.3 6.8 7.9 8.4 6.2 7.8 9.1 6.7
8.8 7.5 7.0 7.3 9.6 5.6 8.0 6.9 7.1 7.9
10.3 7.4 4.9 7.3 8.1 6.2 9.9 5.7 8.6 7.4
8.2 10.8 6.8 7.4 8.9 5.9 7.2 7.9 8.0 6.6

Hypotheses: Let $\mu_{NS}$ denote the population mean for the nonsmokers, and let $\mu_{S}$ denote the population mean for the smokers. We want to test $H_{0}:\mu_{NS} \le \mu_{S}$ vs. $H_{1}:\mu_{NS} \gt \mu_{S}$. Note that this is equivalent to testing $H_{0}:\Delta \le 0$ vs. $H_{1}:\Delta\gt$, where $\Delta=\mu_{NS}-\mu_{S}$.

6.
Turn off ``By Groups''. Perform Welch's approximate t-test using a 1-sided significance level of $\alpha=0.05$.

7.
Calculate the 2-sided 0.95-level confidence interval for $\Delta$. Don't forget that the degrees of freedom are different for this data set, hence you will have to recalculate t0.975,df using the new df. NOTE: You will need to use the formula on the first page of the lab to calculate the confidence interval.

8.
What can be inferred about the birth weights of smoking and nonsmoking mothers?



 
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Dennis Cox
3/26/1998