Respuesta :
Answer:
Confidence interval : [tex]21.506[/tex] to [tex]24.493[/tex]
Step-by-step explanation:
A quality analyst selects twenty racquets and obtains the following lengths:
21, 25, 23, 22, 24, 21, 25, 21, 23, 26, 21, 24, 22, 24, 23, 21, 21, 26, 23, 24
So, sample size = n =20
Now we are supposed to find Construct a 99.9% confidence interval for the mean length of all the junior's tennis racquets manufactured at this plant.
Since n < 30
So we will use t-distribution
Confidence level = 99.9%
Significance level = α = 0.001
Now calculate the sample mean
X=21, 25, 23, 22, 24, 21, 25, 21, 23, 26, 21, 24, 22, 24, 23, 21, 21, 26, 23, 24
Sample mean = [tex]\bar{x}=\frac{\sum x}{n}[/tex]
Sample mean = [tex]\bar{x}=\frac{21+25+23+22+24+21+25+21+23+ 26+ 21+24+22+ 24+23+21+ 21+ 26+23+ 24}{20}[/tex]
Sample mean = [tex]\bar{x}=23[/tex]
Sample standard deviation = [tex]\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}[/tex]
Sample standard deviation = [tex]\sqrt{\frac{(21-23)^2+(25-23)^2+(23-23)^2+(22-23)^2+(24-23)^2+(21-23)^2+(25-23)^2+(21-23)^2+(23-23)^2+(26-23)^2+(21-23)^2+(24-23)^2+(22-23)^2+(24-23)^2+(23-23)^2+(21-23)^2+(21-23)^2+(26-23)^2+(23-23)^2+(24-23)^2}{20-1}}[/tex]
Sample standard deviation= s = [tex]1.72[/tex]
Degree of freedom = n-1 = 20-1 -19
Critical value of t using the t-distribution table [tex]t_{\frac{\alpha}{2}[/tex] = 3.883
Formula of confidence interval : [tex]\bar{x} \pm t_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}}[/tex]
Substitute the values in the formula
Confidence interval : [tex]23 \pm 1.73 \times \frac{1.72}{\sqrt{20}}[/tex]
Confidence interval : [tex]23 -3.883 \times \frac{1.72}{\sqrt{20}}[/tex] to [tex]23 + 3.883 \times \frac{1.72}{\sqrt{20}}[/tex]
Confidence interval : [tex]21.506[/tex] to [tex]24.493[/tex]
Hence Confidence interval : [tex]21.506[/tex] to [tex]24.493[/tex]
Answer:
The 99.9% confidence interval for the mean length of all the junior's tennis racquets manufactured at this plant is [21.7958,24.2042].
Step-by-step explanation:
The given data set is
21, 25, 23, 22, 24, 21, 25, 21, 23, 26, 21, 24, 22, 24, 23, 21, 21, 26, 23, 24
Mean of the data set is
[tex]Mean=\frac{\sum x}{n}[/tex]
[tex]\overline{x}=\frac{21+25+23+22+24+21+25+21+23+26+21+24+22+24+23+21+21+26+23+24}{20}[/tex]
[tex]\overline{x}=23[/tex]
Sample mean of the data is 23.
Standard deviation of population is
[tex]\sigma=\sqrt{\frac{\sum (x-\overline{x})^2}{n}}[/tex]
[tex]\sigma=\sqrt{\frac{56}{20}}[/tex]
[tex]\sigma\approx 1.673[/tex]
Assume that lengths of tennis racquets are normally distributed.
The value of z is 3.219 at 99.9% confidence interval .
We need to find the 99.9% confidence interval for the mean length of all the junior's tennis racquets manufactured at this plant.
[tex]Interval=\overline{x}\pm z*\cdot \frac{\sigma}{\sqrt{n}}[/tex]
[tex]Interval=23\pm 3.219\cdot \frac{1.673}{\sqrt{20}}[/tex]
[tex]Interval=23\pm 1.2042[/tex]
[tex]Interval=[23-1.2042,23+1.2042][/tex]
[tex]Interval=[21.7958,24.2042][/tex]
Therefore the 99.9% confidence interval for the mean length of all the junior's tennis racquets manufactured at this plant is [21.7958,24.2042].
