Question

homework
question 1, 9.3.6 - t
part 2 of 6
hw score: 23.75%, 1.9 of 8 points
points: 0 of 1
listed below are the numbers of words spoken in a day by each member of eight different randomly selected couples. complete parts (a) and (b) below
male: 15,178; 27,485; 14,409; 7,058; 18,497; 15,294; 13,960; 26,335
female: 25,243; 13,139; 17,986; 17,387; 12,678; 16,310; 16,744; 18,183
a. use a 0.05 significance level to test the claim that among couples, males speak fewer words in a day than females.
in this example, μ_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the words spoken by the male minus words spoken by the female. what are the null and alternative hypotheses for the hypothesis test?
h_0: μ_d = 0 word(s)
h_1: μ_d < 0 word(s)
(type integers or decimals. do not round.)
identify the test statistic.
t = (round to two decimal places as needed.)

Explanation:

Step1: Calculate differences (d)

For each couple, \( d = \text{Male} - \text{Female} \):
\( 15178 - 25243 = -10065 \)
\( 27485 - 13139 = 14346 \)
\( 1409 - 17986 = -16577 \)
\( 7058 - 17387 = -10329 \)
\( 18497 - 12678 = 5819 \)
\( 15294 - 16310 = -1016 \)
\( 13960 - 16744 = -2784 \)
\( 26335 - 18183 = 8152 \)

Step2: Find mean of d (\(\bar{d}\))

Sum of \( d \): \( -10065 + 14346 - 16577 - 10329 + 5819 - 1016 - 2784 + 8152 = -12454 \)
\( \bar{d} = \frac{-12454}{8} = -1556.75 \)

Step3: Find standard deviation of d (\(s_d\))

Calculate squared deviations from \(\bar{d}\), sum them, divide by \(n - 1 = 7\), then take square root.
Squared deviations:
\((-10065 + 1556.75)^2 \approx 72674476.56\)
\((14346 + 1556.75)^2 \approx 25234476.56\)
\((-16577 + 1556.75)^2 \approx 22534476.56\)
\((-10329 + 1556.75)^2 \approx 7604476.56\)
\((5819 + 1556.75)^2 \approx 5394476.56\)
\((-1016 + 1556.75)^2 \approx 294476.56\)
\((-2784 + 1556.75)^2 \approx 149476.56\)
\((8152 + 1556.75)^2 \approx 9414476.56\)
Sum of squared deviations: \(72674476.56 + 25234476.56 + 22534476.56 + 7604476.56 + 5394476.56 + 294476.56 + 149476.56 + 9414476.56 = 143300392.48\)
\(s_d = \sqrt{\frac{143300392.48}{7}} \approx \sqrt{20471484.64} \approx 4524.54\)

Step4: Calculate t-statistic

Formula: \( t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}} \), where \(\mu_d = 0\) (from \(H_0\))
\( t = \frac{-1556.75 - 0}{4524.54 / \sqrt{8}} \approx \frac{-1556.75}{1599.23} \approx -0.97 \)

Answer:

\( t \approx -0.97 \)