Respuesta :
Answer:
The 95% confidence interval for the mean is (3.249, 4.324).
We can predict with 95% confidence that the next trial of the paint will be within 3.249 and 4.324.
Step-by-step explanation:
We have to calculate a 95% confidence interval for the mean.
As the population standard deviation is not known, we will use the sample standard deviation as an estimation.
The sample mean is:
[tex]M=\dfrac{1}{15}\sum_{i=1}^{15}(3.4+2.5+4.8+2.9+3.6+2.8+3.3+5.6+3.7+2.8+4.4+4+5.2+3+4.8)\\\\\\ M=\dfrac{56.8}{15}=3.787[/tex]
The sample standard deviation is:
[tex]s=\sqrt{\dfrac{1}{(n-1)}\sum_{i=1}^{15}(x_i-M)^2}\\\\\\s=\sqrt{\dfrac{1}{14}\cdot [(3.4-(3.787))^2+(2.5-(3.787))^2+(4.8-(3.787))^2+...+(4.8-(3.787))^2]}\\\\\\[/tex][tex]s=\sqrt{\dfrac{1}{14}\cdot [(0.15)+(1.66)+(1.03)+...+(1.03)]}[/tex]
[tex]s=\sqrt{\dfrac{13.197}{14}}=\sqrt{0.9427}\\\\\\s=0.971[/tex]
We have to calculate a 95% confidence interval for the mean.
The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.
The sample mean is M=3.787.
The sample size is N=15.
When σ is not known, s divided by the square root of N is used as an estimate of σM:
[tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{0.971}{\sqrt{15}}=\dfrac{0.971}{3.873}=0.2507[/tex]
The t-value for a 95% confidence interval is t=2.145.
The margin of error (MOE) can be calculated as:
[tex]MOE=t\cdot s_M=2.145 \cdot 0.2507=0.538[/tex]
Then, the lower and upper bounds of the confidence interval are:
[tex]LL=M-t \cdot s_M = 3.787-0.538=3.249\\\\UL=M+t \cdot s_M = 3.787+0.538=4.324[/tex]
The 95% confidence interval for the mean is (3.249, 4.324).
