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Answer:
[tex]0.81-2.62\frac{0.34}{\sqrt{110}}=0.725[/tex]
[tex]0.81+2.62\frac{0.34}{\sqrt{110}}=0.895[/tex]
So on this case the 99% confidence interval would be given by (0.725;0.895)
We are confident 99% that the true mean for the echo duration is between 0.725 and 0.895 seconds
Step-by-step explanation:
Previous concepts
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".
[tex]\bar X= 0.81[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=0.34 represent the sample standard deviation
n =110 represent the sample size
Solution to the problem
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=110-1=109[/tex]
Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,109)".And we see that [tex]t_{\alpha/2}=2.62[/tex]
Now we have everything in order to replace into formula (1):
[tex]0.81-2.62\frac{0.34}{\sqrt{110}}=0.725[/tex]
[tex]0.81+2.62\frac{0.34}{\sqrt{110}}=0.895[/tex]
So on this case the 99% confidence interval would be given by (0.725;0.895)
We are confident 99% that the true mean for the echo duration is between 0.725 and 0.895 seconds
The 99% (two-sided) confidence interval for the true average echo duration μ, and the interpretation are;
CI = (0.7265, 0.8935)
Thus means that we are 99% confident that the true mean for the echo duration is between 0.7265 and 0.8935 seconds.
We are given;
Sample size; n = 110
Sample mean; x' = 0.81
Confidence level; CL = 99%
Standard deviation; σ = 0.34
The formula for confidence interval is;
CI = x' ± z(σ/√n)
Where;
x' is sample mean
z is critical value at confidence interval
σ is standard deviation
n is sample size
z at confidence level of 99% is 2.576. Thus;
CI = 0.81 ± 2.576(0.34/√110)
CI = 0.81 ± 0.0835
CI = (0.81 - 0.0835), (0.81 + 0.0835)
CI = (0.7265, 0.8935)
Read more about confidence interval at; https://brainly.com/question/17097944
