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:
![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](https://tex.z-dn.net/?f=M%3D%5Cdfrac%7B1%7D%7B15%7D%5Csum_%7Bi%3D1%7D%5E%7B15%7D%283.4%2B2.5%2B4.8%2B2.9%2B3.6%2B2.8%2B3.3%2B5.6%2B3.7%2B2.8%2B4.4%2B4%2B5.2%2B3%2B4.8%29%5C%5C%5C%5C%5C%5C%20M%3D%5Cdfrac%7B56.8%7D%7B15%7D%3D3.787)
The sample standard deviation is:
![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]}\\\\\\](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cdfrac%7B1%7D%7B%28n-1%29%7D%5Csum_%7Bi%3D1%7D%5E%7B15%7D%28x_i-M%29%5E2%7D%5C%5C%5C%5C%5C%5Cs%3D%5Csqrt%7B%5Cdfrac%7B1%7D%7B14%7D%5Ccdot%20%5B%283.4-%283.787%29%29%5E2%2B%282.5-%283.787%29%29%5E2%2B%284.8-%283.787%29%29%5E2%2B...%2B%284.8-%283.787%29%29%5E2%5D%7D%5C%5C%5C%5C%5C%5C)
![s=\sqrt{\dfrac{1}{14}\cdot [(0.15)+(1.66)+(1.03)+...+(1.03)]}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cdfrac%7B1%7D%7B14%7D%5Ccdot%20%5B%280.15%29%2B%281.66%29%2B%281.03%29%2B...%2B%281.03%29%5D%7D)
![s=\sqrt{\dfrac{13.197}{14}}=\sqrt{0.9427}\\\\\\s=0.971](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cdfrac%7B13.197%7D%7B14%7D%7D%3D%5Csqrt%7B0.9427%7D%5C%5C%5C%5C%5C%5Cs%3D0.971)
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:
The t-value for a 95% confidence interval is t=2.145.
The margin of error (MOE) can be calculated as:
![MOE=t\cdot s_M=2.145 \cdot 0.2507=0.538](https://tex.z-dn.net/?f=MOE%3Dt%5Ccdot%20s_M%3D2.145%20%5Ccdot%200.2507%3D0.538)
Then, the lower and upper bounds of the confidence interval are:
The 95% confidence interval for the mean is (3.249, 4.324).