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Calculating confidence interval

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In an effort to estimate the mean amount spent per household per month in a small Canadian town, data were collected for a sample of 50 households. The sample showed an average amount of $2000 and a standard deviation of $500.

a.
Develop a 90% confidence interval estimate of the population mean amount spent.

b.
Develop a 95% confidence interval estimate of the population mean amount spent.

c.
Discuss what happens to the width of the confidence interval as the confidence level is increased. Does this seem reasonable? Explain.

d.
If the data were collected for a sample of 20 households rather than 50, develop a 95% confidence interval estimate of the population mean amount spent, assuming the population has a normal probability distribution.

e.
Discuss what happens to the width of the confidence interval as the sample size is decreased. Does this seem reasonable? Explain.

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Solution Preview

Hello,

The formula would be: C. I. = M ± (z * SE)
Where:
m=mean
z = the degree of alpha - we get it from a z-table: http://www.statsoft.com/textbook/sttable.html#z
SE = standard error

to get SE = standard deviation / square root of sample size

A. 90% confidence interval = our z score would be 1.65

our SE would be: 500/sq root 50
= 500/7.07
=70.72

Confidence interval = 2000 ± (70.72* 1.65)
Lower level = 2000 - 116.68 = 1883.31
Upper level = 2000 + 116.68 = 2116.68

b) 95% confidence ...

Solution Summary

This problem examines the math behind confidence intervals by looking at several mathematical cases. It gives a full explanation of the results.

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Calculating confidence interval for proportions

An interactive poll found out that 330 adults of 2,396 adults over the age of 18 have at least 1 tattoo.
A) Obtain a point estimate of the proportion of adults that have at least 1 tattoo. p̂=____?

B) Construct a 90% confidence interval for the proportion of adults that have at least 1 tattoo. Select the correct choice below and if necessary fill in the blanks
a) Lower bound ___________ Upper bound ________________ round to three decimal places
b) The requirements for conducting a confidence interval are not satisfied.

C) Construct a 99% confidence interval for the proportion of adults that have at least 1 tattoo. Select the correct choice below and if necessary fill in the blanks
a) Lower bound ___________ Upper bound ________________ round to three decimal places
b) The requirements for conducting a confidence interval are not satisfied.

D) Choose the correct answer below
a) Increasing the level of confidence has no effect on the interval
b) Increasing the level of confidence narrows the interval
c) Increasing the level of confidence widens the interval
d) It is not possible to tell the effect of increasing the level of interval of confidence width of the interval since
the requirements for constructing a confidence interval in parts a and b were not met.

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