Women stereotypically talk more than men do and researchers wondered how much mo

Question

Women stereotypically talk more than men do and researchers wondered how much more. Suppose a study attempted to determine the difference in the mean number of words spoken by men or women per day. The results of the study are summarized in the table. PopulationSample Sample S sandard Standard error Groupmean size nw-42 Xw-14297 nm=37Fm=14060 deviation Sw = 6441 sm=9065 estimate SEw-994 SEm=1490 mean women w (unknown) menHm (unknown) df = 64.0237 Assume the conditions are satisfied for a two-sample 1-confidence interval. First, determine the positive critical value, f, for a 90% confidence interval to estimate how many more words women speak each day on average compared to men Hm. Give your answer precise to at least three decimal places. Scroll

Explanation / Answer

PART A.
Given that,
mean(x)=14297
standard deviation , s.d1=6441
number(n1)=42
y(mean)=14060
standard deviation, s.d2 =9065
number(n2)=37
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, = 0.1
from standard normal table, two tailed t /2 =1.688
since our test is two-tailed
reject Ho, if to < -1.688 OR if to > 1.688
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =14297-14060/sqrt((41486481/42)+(82174225/37))
to =0.1323
| to | =0.1323
critical value
the value of |t | with min (n1-1, n2-1) i.e 36 d.f is 1.688
we got |to| = 0.13231 & | t | = 1.688
make decision
hence value of |to | < | t | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 0.1323 ) = 0.895
hence value of p0.1 < 0.895,here we do not reject Ho
ANSWER:
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test statistic(t): 0.1323

PART B.
CI = x1 - x2 ± t a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
where,
x1,x2 = mean of populations
sd1,sd2 = standard deviations
n1,n2 = size of both
a = 1 - (confidence Level/100)
ta/2 = t-table value
CI = confidence interval
CI = [( 14297-14060) ± t a/2 * sqrt((41486481/42)+(82174225/37)]
= [ (237) ± t a/2 * 1791.284]
= [-2786.687 , 3260.687]
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interpretations:
1. we are 90% sure that the interval [-2786.687 , 3260.687] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population proportion
Lower limit:-2786.687
upper limit:3260.687

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