Alternative hypothesis, two sample t statistic, degree of freedom

Question

Equip male and female students with a small device that secretly records sound for a random 30 seconds during each 12.5-minute period over two days. Count the words each subject speaks during each recording period, and from this, estimate how many words per day each subject speaks. The published report includes a table summarizing six such studies. Here are two of the six:

       Sample Size  Estimated Average Number

                                  (SD) of Words Spoken per Day

Study	Women	Men	Women	Men
1	55	55	15523 (7838)	16893 (9274)
2	27	20	16383 (7980)	12840 (8004)

Readers are expected to understand this to mean, for example, the 55 women in the first study had x⎯⎯⎯�¯ = 15523 and s = 7838.

It is commonly thought that women talk more than men. Does either of the two samples support this idea? For each study:

(a) State the alternative hypothesis in terms of the population mean number of words spoken per day for men (μM)(��) and for women (μF)(��). If necessary, use != to represent ≠≠.

Study 1: μM��  μF��

Study 2: μM��  μF��

(b) Find the two-sample t statistic to test μF−μM��−��.

Study 1: 

Study 2: 

(c) What degrees of freedom does Option 2 use to get a conservative P-value?

Study 1: 

Study 2: 

(d) Compare your value of t with the critical values in Table C. What can you say about the P- value of the test?

Study 1: P- value >> 

Study 2:  ≤≤ P- value ≤≤ 

Answer 1

Hi Lauren,

I noticed you've posted a few similar questions recently. I actually recognize these questions, but I don't recall the source (either Triola's book, OpenStax, or WeBWorK). I'm new to Wyzant and can't speak to their policies on copy-pasting from other sources, but the formatting errors are a clear indication that this is what you're doing here. Just letting you know.

I will at least answer Part C for you. "Option 2" or the "Conservative Option" refer to a particular method for computing a T-critical value for Two-Sample Hypothesis Tests when comparing population means, assuming the population's standard deviations are unknown (the method is also used for Confidence Intervals for the Difference in Means). You'll need Option 2 when using a table of critical values, because such tables require Degrees of Freedom to be a whole number. The computation with this method is very simple; take the SMALLER of the two samples sizes and subtract 1 to get your df. In your case:

1. Study 1 has both sample sizes of 55, so Option 2 uses df = 55 - 1 = 54.
2. Study 2 has sample sizes of 27 and 20, so Option 2 uses df = 20 - 1 = 19.

When you get to part D, you might be using a table that doesn't have df = 54 (the ones I've had to teach from start skipping values around df = 30). In such a case, always use the next larger df in the table (probably 60).