Respuesta :
Answer:
a) [tex]0.995 - 2.58 \frac{0.02}{\sqrt{50}}=0.988[/tex]
[tex]0.995 + 2.58 \frac{0.02}{\sqrt{50}}=1.00[/tex]
The 99% confidence interval is given by (0.988;1.00)
b) NO, because a 1-gallon of bottle contain exactly 1-gallon of water and this value is inside of the 99% confidence interval.
c) Yes we are assuming this (normality) in order to construct the confidence interval.
d) [tex]0.995 - 1.96 \frac{0.02}{\sqrt{50}}=0.989[/tex]
[tex]0.995 + 1.96 \frac{0.02}{\sqrt{50}}=1.00[/tex]
The 95% confidence interval is given by (0.989;1.00)
The confidence interval not changes too much, we got very similar answers.
Step-by-step explanation:
1) Notation and definitions
n=50 represent the sample size
[tex]\bar X=0.995[/tex] represent the sample mean
[tex]s=0.02[/tex] represent the sample standard deviation
m represent the margin of error
Confidence =99% or 0.99
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".
Part a
In order to find the critical value is important to mention that we know about the population standard deviation, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex].
We can find the critical values in excel using the following formulas:
"=NORM.INV(0.005,0,1)" for [tex]z_{\alpha/2}=-2.58[/tex]
"=NORM.INV(1-0.005,0,1)" for [tex]z_{1-\alpha/2}=2.58[/tex]
The critical value [tex]zc=\pm 2.58[/tex]
The interval for the mean is given by this formula:
[tex]\bar X \pm z_{c} \frac{\sigma}{\sqrt{n}}[/tex]
And calculating the limits we got:
[tex]0.995 - 2.58 \frac{0.02}{\sqrt{50}}=0.988[/tex]
[tex]0.995 + 2.58 \frac{0.02}{\sqrt{50}}=1.00[/tex]
The 99% confidence interval is given by (0.988;1.00)
Part b
NO, because a 1-gallon of bottle contain exactly 1-gallon of water and this value is inside of the 99% confidence interval.
Part c
Yes we are assuming this (normality) in order to construct the confidence interval.
Part d
In order to find the critical value is important to mention that we know about the population standard deviation, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex].
We can find the critical values in excel using the following formulas:
"=NORM.INV(0.025,0,1)" for [tex]z_{\alpha/2}=-1.96[/tex]
"=NORM.INV(1-0.025,0,1)" for [tex]z_{1-\alpha/2}=1.96[/tex]
[tex]0.995 - 1.96 \frac{0.02}{\sqrt{50}}=0.989[/tex]
[tex]0.995 + 1.96 \frac{0.02}{\sqrt{50}}=1.00[/tex]
The 95% confidence interval is given by (0.989;1.00)
The confidence interval not changes too much, we got very similar answers.
Refer the below solution for better understanding.
Step-by-step explanation:
Given :
Sample size, n = 50
Sample mean, [tex]\bar{X} = 0.995[/tex]
Sample standard deviation, s = 0.02
Confidence - 99% = 0.99
Calculation :
Let m represent the margin of error.
a) Significance level is given by,
[tex]\alpha =1-0.99=0.01 \;and\;\dfrac{\alpha}{2}=0.005[/tex]
We can find the critical values by,
[tex]z_\frac{\alpha }{2}=-2.58[/tex]
[tex]z_1_-_\frac{\alpha }{2}=2.58[/tex]
[tex]z_c= \pm2.58[/tex]
Interval for the mean is,
[tex]\bar{X} \pm z_c\dfrac{\sigma}{\sqrt{n} }[/tex]
[tex]0.995-2.58\dfrac{0.02}{\sqrt{50} }=0.988[/tex]
[tex]0.995+2.58\dfrac{0.02}{\sqrt{50} }=1.00[/tex]
99% confidence interval is given by (0.988,1.00).
b) NO, because one gallon of bottle contain exactly one gallon of water and this value is inside of the 99% confidence interval.
c) Yes we assume that the population amount of water per bottle is normally distributed here in order to construct the confidence interval.
d) Significance level iis given by,
[tex]\alpha =1-0.95=0.05 \;and\;\dfrac{\alpha}{2}=0.025[/tex]
We can find the critical values by,
[tex]z_\frac{\alpha }{2}=-1.96[/tex]
[tex]z_1_-_\frac{\alpha }{2}=1.96[/tex]
[tex]z_c= \pm1.96[/tex]
Interval for the mean is,
[tex]\bar{X} \pm z_c\dfrac{\sigma}{\sqrt{n} }[/tex]
[tex]0.995-1.96\dfrac{0.02}{\sqrt{50} }=0.989[/tex]
[tex]0.995+1.96\dfrac{0.02}{\sqrt{50} }=1.00[/tex]
95% confidence interval is given by (0.989,1.00).
The confidence interval not changes too much, we got very similar answers.
For more information, refer the link given below
https://brainly.com/question/14747159?referrer=searchResults
