Question 1202936: The owner of Britten’s Egg Farm wants to estimate the mean number of eggs produced per chicken. A sample of 15 chickens shows they produced an average of 22 eggs per month with a standard deviation of 7 eggs per month.
What is the value of the population mean
What is the best estimate of this value?
For a 90% confidence interval, what is the value of t?
What is the margin of error?
Develop the 90% confidence interval for the population mean
Found 2 solutions by Theo, math_tutor2020:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample mean is 22
sample standard deviation is 7
sample size is 15
population mean is assumed to be 22 plus or minus a margin of error.
90% two tailed confidience interval with 14 degrees of freedom leads to a critical t-score of plus or minus 1.7613.
at 1.7613 t-score, the t-score formula becomes 1.7613 = (x - 22) / s, where s is the standard error.
s = standard deviation of sample divided by square root of sample size = 7 / sqrt(15) = 1.80739.
when t = 1.7613, t-score formula becomes 1.7613 = (x - 22) / 1.80739.
solve for x to get x = 25.183356.
round to 4 decimal places = 25.1834.
when t = -1.7613, t-score formula becomes -1.7613 = (x - 22) / 1.80739.
solve for x to get x = 18.81664.
round to 4 decimal places = 18.8166
your 90% confidence interval indicates the true population mean will be somewhere between 18.8166 and 25.1834.
the margin of error is equal to the mean plus or minus (25.1834 - 22) = 3.1834
22 + 3.1834 = 25.1834
22 - 3.1834 = 18.8166
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
mu = population mean
sigma = population standard deviation
These two values are unknown.
xbar = sample mean = 22
s = sample standard deviation = 7
The value of mu is unknown.
Often we won't ever know this value unless doing a census, but such a task is very costly in most cases.
The best estimate for mu is xbar = 22.
n = sample size
n = 15
df = degrees of freedom
df = n-1
df = 15-1
df = 14
I'll use this table
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
to find the critical t value.
A similar table is found in the back of your stats textbook.
A stats calculator is another option.
Using that table, look at the row labeled "df = 14" and the column labeled "confidence level = 90%"
The confidence level labels are at the bottom of that table.
The row and column mentioned has this value
t = 1.761
It indicates that:
P(-1.761 < t < 1.761) = 0.90 approximately when df = 14.
That critical t value, along with the standard deviation and sample size, will be useful to compute the margin of error
E = margin of error
E = t*s/sqrt(n)
E = 1.761*7/sqrt(15)
E = 3.18281771391326
E = 3.182818
This result is approximate.
From here we use xbar and the margin of error to find the lower (L) and upper (U) boundaries of the confidence interval (L,U)
L = lower boundary
L = xbar - E
L = 22 - 3.182818
L = 18.817182
L = 18.8
and
U = upper boundary
U = xbar + E
U = 22 + 3.182818
U = 25.182818
U = 25.2
Both of these results are approximate.
The 90% confidence interval estimating mu is approximately 18.8 < mu < 25.2
It is of the form L < mu < U
We are 90% confident that mu is somwehere between 18.8 and 25.2
The notation 18.8 < mu < 25.2 condenses down to (18.8, 25.2) to be written in the form (L,U)
This second type of notation is most common in stats and other settings.
The drawback is that we lose context of what parameter we're trying to estimate.
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Summary
What is the value of the population mean?
Unknown
What is the best estimate of this value?
xbar = sample mean = 22 eggs per month
For a 90% confidence interval, what is the value of t?
approximately t = 1.761
What is the margin of error?
approximately 3.182818
Develop the 90% confidence interval for the population mean
(18.8, 25.2)
The confidence interval boundaries are approximate because the margin of error value is approximate.
Round each value however your teacher instructs.
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