any1 who can do as much of this as possible!??
6.Assuming that the following sample of 12 numbers arise from a Normal
distribution with mean and standard deviation σ = 5.2, obtain 95%, 99%
and 99.9% conﬁdence intervals for .
92.3 98.4 102.1 100.5 105.3 96.5
96.2 105.3 92.7 107.1 96.9 89.2
7. For the data in Q6, evaluate the probability or p-value for testing the null
hypothesis H0 : = 100 against the two-sided alternative hypothesis
H1 : = 100. What evidence do the CIs in Q6 and this p-value oﬀer in
support of H0?
8.A researcher in Human Resources Management wants to determine the
mean monthly salary of bank managers, and has a budget of £900 to
conduct a survey and construct a conﬁdence interval based on survey
responses. The researcher knows that the population standard deviation of
monthly salary σ is £1000, and allows £1.50 to cover the cost of contacting
each respondent. If the error in estimating the mean is to be less than
£100 with a 95% level of conﬁdence, show that the required sample size
is within budget.
• Is the budget suﬃcient if the researcher wishes to work with a 99%
level of conﬁdence?
• What budget is required if the researcher wishes to retain a 95% level
of conﬁdence, but wishes to reduce the associated error to £50?
9. An initial sample of size 25 from a Normal population with mean and
standard deviation σ = 10 gives
¯X = 54.3. Show that the test-statistic for
testing the null hypothesis H0 : = 50 against the two-sided alternative
hypothesis H1 : = 50 leads to a p-value less than 0.05.
A further sample of size 75 is then taken from the same population, and it
is proposed to reconsider the same hypothesis using the combined sample
of 100 values.
• What values of the mean of these 100 values will lead to p-value
greater than 0.05 when testing H0 for the combined sample?
• Consider the further sample: determine the values of the sample
mean in this further sample which will make the ﬁnal p-value greater
than 0.05, and ﬁnd the probability of obtaining a sample mean for
the further sample in this range.
10. A sample of size n = 40 is taken from a N
, 10
2
population, and the
data analysed by a consultant who, without taking statistical advice, has
decided to reject H0 : = 10 if ¯X < 8 or ¯X > 12.
• What is the probability that this rule results in the consultant making
a Type I error?
• Having taking some statistical advice, the consultant decides to change
the rule to one where the null hypothesis will be rejected if ¯X < 10−c
or ¯X > 10 + c. What value should c take to make the probability of
a Type I error equal to 0.05?
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