The following data lists the ages of a random selection of actresses when they won an award in the category of Best Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts (a) and (b) below.
Actress left parenthesis years right parenthesis |
---|
29
25
34
27
38
26
26
45
27
36
Actor left parenthesis years right parenthesis |
---|
62
36
39
38
31
33
52
40
41
40
a. Use the sample data with a 0.05 significance level to test the claim that for the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0 (indicating that the Best Actresses are generally younger than Best Actors).
In this example,mu Subscript 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 actress's age minus the actor's age. What are the null and alternative hypotheses for the hypothesis test?
Upper H 0: mu Subscript d
▼
greater than
equals
not equals
less than ________year(s)
Upper H 1: mu Subscript d
▼
equals
greater than
not equals
less than _____year(s)
(Type integers or decimals. Do not round.)
Identify the test statistic.
t=________
(Round to two decimal places as needed.)
Identify the P-value.
P-value=_______
(Round to three decimal places as needed.)
What is the conclusion based on the hypothesis test?
Since the P-value is :
▼
greater than
less than or equal to the significance level,
▼
reject
fail to reject
the null hypothesis. There
▼
is not
is
sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.
b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
The confidence interval is ______year(s)less than mu Subscript d less than_________year(s).
(Round to one decimal place as needed.)
What feature of the confidence interval leads to the same conclusion reached in part (a)?
What feature of the confidence interval leads to the same conclusion reached in part (a)?
Since the confidence interval contains:
▼only negative numbers,
zero,
only positive numbers,
▼fail to reject
reject
the null hypothesis.
Ho :µd=0
Ha :µd <0
Sample #1 | Sample #2 | difference , Di =sample1-sample2 | (Di - Dbar)² |
29 | 62 | -33 | 533.610 |
25 | 36 | -11 | 1.210 |
34 | 39 | -5 | 24.010 |
27 | 38 | -11 | 1.210 |
38 | 31 | 7 | 285.610 |
26 | 33 | -7 | 8.410 |
26 | 52 | -26 | 259.210 |
45 | 40 | 5 | 222.010 |
27 | 41 | -14 | 16.810 |
36 | 40 | -4 | 34.81 |
sample 1 | sample 2 | Di | (Di - Dbar)² | |
sum = | 313 | 412 | -99 | 1386.900 |
mean= | 31.3000 | 41.2000 | -9.90000 |
Level of Significance , α = 0.05
sample size , n = 10
mean of sample 1, x̅1=31.3000
mean of sample 2, x̅2=41.2000
mean of difference , D̅ =-9.9000
std dev of difference , Sd = √ [ (Di-Dbar)²/(n-1) = 12.4137
std error , SE = Sd / √n = 3.9256
t-statistic = (D̅ - µd)/SE = -2.52
Degree of freedom, DF=n - 1 = 9
p-value = 0.016 [excel function: =t.dist(t-stat,df) ]
Conclusion: p-value <α , Reject null hypothesis
Since the P-value is
less than or equal to the significance level,
reject
the null hypothesis. There
is
sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.
--------------------------------------------
b)
Degree of freedom, DF=n - 1 = 9
t-critical value = t α/2,df = 2.2622 [excel function: =t.inv.2t(α/2,df) ]
std dev of difference , Sd = 12.4137
std error , SE = Sd / √n = 3.9256
margin of error, E = t*SE = 8.880228
mean of difference , D̅ =-9.9000
confidence interval is
Interval Lower Limit=D̅ - E =-18.7802
Interval Upper Limit=D̅ + E =-1.0198
so, confidence interval is (-18.8< Dbar < - 1.0)
Since the confidence interval contains:only negative numbers,
reject
the null hypothesis.
The following data lists the ages of a random selection of actresses when they won an award in th...
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