Question

A statistic instructor randomly selected four bags of oranges, each bag labeled 10 pounds, and weighed the bags.They weighed 9.3, 9.7, 9.2, and 9.7 pounds. assume that the distribution of the weights is normal. Find a 95% confidence interval for the mean weight of all bags of oranges.
We are 95% confident the population mean is between ____ and ____?

Answer 1

Answer:

95% confident the population mean is between (9.1,9.9)

Step-by-step explanation:

We are given the following data set:

9.3, 9.7, 9.2, 9.7

Formula:

$\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}$  

where $x_i$ are data points, $\bar{x}$ is the mean and n is the number of observations.  

$Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}$

$Mean =\displaystyle\frac{37.9}{4} = 9.475$

Sum of squares of differences = 0.2075

$S.D = \sqrt{\frac{0.2075}{3}} = 0.262$

95% Confidence interval:  

$\bar{x} \pm t_{critical}\displaystyle\frac{s}{\sqrt{n}}$  

Putting the values, we get,  

$t_{critical}\text{ at degree of freedom 3 and}~\alpha_{0.05} = \pm 3.182$  

$9.475 \pm 3.182(\frac{0.262}{\sqrt{4}} ) = 9.475 \pm 0.416842 =(9.058158,9.891842) \approx (9.1,9.9)$