Question

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	female
25,783	13,860
1,434	18,267
7,915	18,193
18,978	12,868
15,485	17,080
13,877	16,238
26,757	19,160

1. use a 0.01 significance level to test the claim that among couples, males speak fewer words in a day than females.

in this example, \\( \mu_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: \mu_d = 0 \\) word(s)
\\( h_1: \mu_d < 0 \\) word(s)
(type integers or decimals. do not round.)
identify the test statistic.
\\( t = \square \\) (round to two decimal places as needed.)

Explanation:

Step1: Calculate differences (d)

For each couple, \( d = \text{Male} - \text{Female} \):

• \( 16100 - 25435 = -9335 \)
• \( 25783 - 13860 = 11923 \)
• \( 1434 - 18267 = -16833 \)
• \( 7915 - 18193 = -10278 \)
• \( 18978 - 12868 = 6110 \)
• \( 15485 - 17080 = -1595 \)
• \( 13877 - 16238 = -2361 \)
• \( 26757 - 19160 = 7597 \)

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

Sum of \( d \): \( -9335 + 11923 - 16833 - 10278 + 6110 - 1595 - 2361 + 7597 = -25772 \)
\( \bar{d} = \frac{-25772}{8} = -3221.5 \)

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

1. Calculate squared differences from mean:

\( (-9335 + 3221.5)^2 = (-6113.5)^2 = 37374982.25 \)
\( (11923 + 3221.5)^2 = (15144.5)^2 = 229355880.25 \)
\( (-16833 + 3221.5)^2 = (-13611.5)^2 = 185273932.25 \)
\( (-10278 + 3221.5)^2 = (-7056.5)^2 = 49794292.25 \)
\( (6110 + 3221.5)^2 = (9331.5)^2 = 87076902.25 \)
\( (-1595 + 3221.5)^2 = (1626.5)^2 = 2645402.25 \)
\( (-2361 + 3221.5)^2 = (860.5)^2 = 740460.25 \)
\( (7597 + 3221.5)^2 = (10818.5)^2 = 117040042.25 \)

1. Sum of squared differences: \( 37374982.25 + 229355880.25 + 185273932.25 + 49794292.25 + 87076902.25 + 2645402.25 + 740460.25 + 117040042.25 = 709301894 \)

1. Variance: \( \frac{709301894}{8 - 1} \approx 101328842 \)
2. \( s_d = \sqrt{101328842} \approx 10066.22 \)

Step4: Calculate t-statistic

Formula: \( t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}} \), where \( \mu_d = 0 \), \( n = 8 \)
\( t = \frac{-3221.5 - 0}{10066.22 / \sqrt{8}} \approx \frac{-3221.5}{3556.44} \approx -0.91 \) (rounded to two decimals)

Answer:

\( t \approx -0.91 \)