Unformatted text preview: ounding variables that are not being controlled for in this study, for
example the gender of the bull frog, leg length, weight, etc.
12. New York is known as “the city that never sleeps”. A random sample of 25 New Yorkers were asked
how much sleep they get per night. A summary of some sample statistics are shown below. Assume
that the distribution of the amount of sleep these New Yorkers get unimodal and symmetric. Based
on this sample, is there signiﬁcant evidence to suggest that New Yorkers on average sleep less than 8
hrs a night?
n
25 x
¯
7.73 s
0.77 (a) Write the hypotheses in symbols and in words.
H0 : µ = 8 (New Yorkers on average sleep 8 hrs per night.)
HA : µ < 8 (New Yorkers on average sleep less than 8 hrs per night.)
(b) Are the assumptions/conditions for inference satisﬁed?
1. Independence Assumption:
• Random Sampling Condition: We are told that the sample is random.
• 10% Condition: 25 < 10% of all New Yorkers.
Since we have a random sample and the 10% condition is satisﬁed, we can assume that the
number of hours one New Yorker in the sample sleeps is independent of another.
2. Nearly Normal Condition: We are told that the distribution of the sample is unimodal and
symmetric, hence we can assume the sample comes from a nearly normal distribution. 8 (c) Calculate the test statistic and ﬁnd the pvalue.
The test statistic can be calculated as follows.
t= x − µ0
¯
s
√
n = 7.73 − 8
0.
√77
25 = −1.75 df = 25 − 1 = 24
0.025 < p − value < 0.05
(d) Interpret the pvalue in context.
If in fact New Yorkers on average sleep 8 hours per night, the probability of getting a random
sample of 25 New Yorkers where the average amount of sleep is 7.73 hrs per night or less is
between 0.025 and 0.05.
(e) Based on the hypothesis test is there signiﬁcant evidence to suggest that New Yorkers on average
sleep less than 8 hrs a night?
Since pvalue < α (use α = 0.05 since not given) we reject the null hypothesis. There is suﬃcient
evidence to suggest that New Yorkers on average sleep less than 8 hours per night.
(f) Would you expect a conﬁdence interval for the population mean of at an equivalent conﬁdence
level as the hypothesis test to include 8? Explain.
No, the hypothesis test suggests that the average amount of sleep New Yorkers get is signiﬁcantly
lower than 8 hours per night therefore we wouldn’t expect 8 to be in the interval.
13. The previous question gives some summary statistics on the number of hours of sleep 25 randomly
sampled New Yorkers get per night.
(a) Calculate a 90% conﬁdence interval for the number of hours of New Yorkers sleep on average and
interpret this interval in context.
Before we can calculate the conﬁdence interval, we need to ﬁnd the degrees of freedom and t∗ .
df
df = 25 − 1 = 24 t∗ = 1.711
df A 90% conﬁdence interval can be calculated as follows:
s
0.77
x ± t∗ √ = 7.73 ± 1.711 √
¯
df
n
25
= 7.73 ± 0.26
= (7.47, 7.99) We are 90% conﬁdent that New Yorkers on...
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This document was uploaded on 12/04/2013.
 Winter '13

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