Respuesta :
Answer:
a) [tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]
b) The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean
Step-by-step explanation:
Part a
The best point of estimate for the population mean is the sample mean given by:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
Since is an unbiased estimator [tex] E(\bar X) = \mu[/tex]
Data given: 2 , 2 , 1 , 3 , 1 , 0 , 4 , 1
So for this case the sample mean would be:
[tex] \bar X = \frac{2+2+1+3+1+0+4+1}{8}= 1.75[/tex]
Part b
Previous concepts
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".
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".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
The margin of error is given by this formula:
[tex] ME=t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (2)
And for this case we know that ME =0.89 with a confidence of 95%
So then the limits for our confidence level are:
[tex] Lower= \bar X -ME= 1.75- 0.89=0.86[/tex]
[tex] Upperr= \bar X +ME= 1.75+0.89=2.64[/tex]
So then the best answer for this case would be:
The margin of error indicates we can be 95%confident that the sample mean falls within 0.89 of the population mean
