Question 1208010: A random sample of n = 36 observations has a mean x = 28.7 and a standard deviation s = 3.8.
(a) Find a 90% lower confidence bound for the population mean 𝜇. (Round your answer to two decimal places.)
(b) How many observations do you need to estimate 𝜇 to within 0.6, with probability equal to 0.95? (Round your answer up to the nearest whole number.)
observations
Answer by math_tutor2020(3816)   (Show Source):
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Answers:
(a) 27.66
(b) 155
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Explanation for Part (a)
n = 36 = sample size
xbar = 28.7 = sample mean
s = 3.8 = sample standard deviation
mu = population mean = unknown
sigma = population standard deviation = unknown
Since n = 36 fits the criteria n > 30, we can use the Z distribution even if we don't know the value of sigma.
At a 90% confidence level, the critical z value is roughly z = 1.645
Use a reference table or stats calculator to determine this.
I'm using this table
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
A table like this should be somewhere in the back of your stats textbook.
Your professor will likely hand out such a table during exams if s/he expects you to use them.
Refer to the bottom row of the table where it lists the confidence levels. Just above "90%" is the value 1.645
E = margin of error for population mean
E = z*s/sqrt(n)
E = 1.645*3.8/sqrt(36)
E = 1.041833333333 approximately
L = lower boundary of confidence interval
L = xbar - E
L = 28.7 - 1.041833333333
L = 27.658166666667
L = 27.66 which is the answer to part (a).
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Explanation for Part (b)
E = margin of error for population mean
E = z*s/sqrt(n)
E*sqrt(n) = z*s
sqrt(n) = z*s/E
n = (z*s/E)^2
This formula gives us the minimum sample size needed when we specify a desired margin of error.
In this case we have E = 0.6
The portion that says "with probability equal to 0.95" refers to a 95% confidence interval.
At 95% confidence, the z critical value is roughly z = 1.960 (use a table or stats calculator).
Here are the inputs we'll need
z = 1.960 (approximate)
s = 3.8
E = 0.6
Let's calculate the minimum sample size.
n = (z*s/E)^2
n = (1.960*3.8/0.6)^2
n = 154.090844444444
n = 155 .... rounding up to nearest integer.
Despite n being much closer to 154, we must round up to 155 to clear the hurdle.
Let's see what happens when we try n = 154.
E = z*s/sqrt(n)
E = 1.960*3.8/sqrt(154)
E = 0.60018 approximately
This is slightly over the 0.6 threshold we want.
The goal is to get E = 0.6 exactly or E < 0.6
Now try n = 155.
E = z*s/sqrt(n)
E = 1.960*3.8/sqrt(155)
E = 0.59824 approximately
We are now under the 0.6 threshold.
Answer to part (b) is 155
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