Answer:
a)For this case the best estimator for the true mean is the sample mean because:
And if we assume that each observation follows a normal distribution
then we have:
So then yes the
c)
Step-by-step explanation:
Previous concepts
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
represent the sample mean
population mean (variable of interest)
represent the population standard deviation
n=50 represent the sample size
Assuming the X follows a normal distribution
The sample mean is distributed on this way:
A. What is the point estimate in this scenario? Is it an unbiased estimator?
For this case the best estimator for the true mean is the sample mean because:
And if we assume that each observation follows a normal distribution
then we have:
So then yes the ,
and
Using the normal standard table, excel or a calculator we see that:
C. What is the margin of error for a 95% interval in this scenario? Show your work.
The margin of error is given by:
If we replace the values that we have we got:
The confidence interval on this case is given by:
(1)
Since we have all the values we can replace:
So on this case the 95% confidence interval would be given by (3.989;4.211)