A random sample of 12 men indicates an average height of 69 inches with a standard deviation of 2.1. A random sample of 8 women indicates an average height of 65.5 inches with a standard deviation of 3.25.
I started by finding the standard error with SE=sqrt((2.1^2/12)+(3.25^2/8)) which gave about SE=1.299
Then I found the t* (because n<30 in both) by using alpha=0.05 with degrees of freedom=(smaller n)-1, 8-1=7 and looked it up on the t table to find t*=2.36
The I used the confidence interval formula [(barX_1 + barX_2 - t(SE) ), (barX_1 + barX_2 + t(SE) )]
to get [((69-65.5)-(2.36)(1.299)), ((69-65.5)+(2.36)(1.299))]
To get the answer to the second part of the question I then
I agree with your math, but I’m confused about the interpretation. Assuming the claim “The average height of women is less than the average height of men”… did you fail to disprove this? Am I interpreting the claim correctly?
I haven’t answered the second part of the question yet
Okay, cool. The test statistic is \frac{69-65.5}{1.299} where 1.299 is your SE and 69 and 65.5 are our average heights of men & women respectively. That’s equal to 2.694. On the z^* table, that corresponds to .9964, but I’m not sure I’m looking in the right place, so I don’t actually know what that means…
I’m still confused about interpretation in general. Suggestions?
I thought we would use the t table but I am not sure and the formula I have for test statistic for 2 means has mew in it so I am confused on what to do
\mu is the same as the x-bar, referring to mean. I think you might be right about the t-table.
Somebody else want to give us a hand here?
I think you’ve got the right test statistic. Now you’ve got to use the t-table to determine if that test statistic is large enough to reject the null hypothesis.
Speaking of the null hypothesis, it would be nice if the hypotheses were clearly stated at the outset.
Correct me if I am wrong, but is our degree of freedom the smallest sample size - 1?
Yes, that is correct!
Okay, I’ll try and put this into the “steps” I’ve been following for all hypothesis test problems.
Write claims. Our claim is that women are shorter than men. Our counter-claim would be that women are just as tall as or taller than men. To use notation, I could write these claims as \mu_w<\mu_m and \mu_w \ge \mu_m respectively.
Now, decide which of those is the null and which is the alternative hypothesis. I think I heard somewhere that a null hypothesis is invariably the one with the equal sign in, which would make it H_0 = \mu_w \ge \mu_m and our alternative hypothesis H_A = \mu_w<\mu_m
Determine an alpha value (given). In this case, you want us to be 95% confident that women are shorter than men; therefore, \alpha = 1-.95 = .05
Calculate the test statistic. Welp, we already did this. No sense in beating a dead horse, we know that our test statistic is 2.694
Draw a picture. Determine if the test statistic is to the left or right of where the alpha value says it ought to be. Looking up our alpha value in the t-table, it says 1.89 for one tail of a graph with 7 degrees of freedom. The picture isn’t strictly speaking necessary, but it helps me get a feel for what’s going on. That’s why I’ve included it in these steps. In this case, I drew this graph and shaded the tail. The t-table value and test statistic are both marked.
Make a decision. If the area in the tail of the graph is less than or equal to what the alpha value says it ought to be, then reject the null hypothesis. If the area in the tail of the graph is greater than what the alpha value says it ought to be, fail to reject the null hypothesis. In the case of this graph, the area is in fact less than what the alpha value says it ought to be. That is to say, if I were to mark the tail starting at the test statistic, I wouldn’t shade the whole region that I shaded marking the tail at the alpha value. So we reject our null hypothesis.
Interpret what that decision means. We just rejected the null hypothesis. What was the null hypothesis? The null hypothesis was that women are as tall as men or taller. So, my interpretation of this whole mess I just went through is that there is enough evidence to support the claim that women are, on average, shorter than men.
How’s that?
