Respuesta :
Answer:
The true statements Include
A) The length of a confidence interval can be determined if you know only the margin of error.
B) The margin of error can be determined if you know only the length of the confidence interval.
D) The confidence interval can be obtained if you know only the margin of error and the sample mean.
G) The margin of error can be determined if you know only the confidence level, population standard deviation, and sample size.
F) The confidence level can be determined if you know only the margin of error, population standard deviation, and sample size.
The false statements include
C) The confidence interval can be obtained if you know only the margin of error.
E) The margin of error can be determined if you know only the confidence level.
F) The confidence level can be determined if you know only the margin of error.
Step-by-step explanation:
Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample mean) ± (Margin of error)
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error of the mean)
The critical value is obtained using the confidence level (and sample size for t-distributions).
Standard Error of the mean = (Standard deviation)/[√(sample size)]
Standard error of the mean = σₓ = (σ/√n)
Taking each of the statements one at a time
a) The length of a confidence interval can be determined if you know only the margin of error.
According to the mathematical expression of the confidence interval, the length of the confidence interval is 2 × Margin of error. Hence, the length of the confidence interval can be truly obtained from only the margin of error.
This statement is true.
b) The margin of error can be determined if you know only the length of the confidence interval.
This is another way of phrasing statement A. The margin of error is half of the length of the confidence interval.
Hence, this statement is also true.
c) The confidence interval can be obtained if you know only the margin of error.
The confidence interval can be obtained from an expression that involves the sample mean and margin of error.
Confidence Interval = (Sample mean) ± (Margin of error)
The confidence interval cannot be obtained from just the margin of error. Hence, this statement is false.
d) The confidence interval can be obtained if you know only the margin of error and the sample mean.
Like o stated in (c) above, the confidence interval can be obtained from an expression that involves the sample mean and margin of error.
Confidence Interval = (Sample mean) ± (Margin of error)
Hence, this statement is true.
e) The margin of error can be determined if you know only the confidence level.
To obtain the margin of error, if the confidence interval isn't used, an expression involving the critical value and the standard error of the mean is used.
Margin of Error = (Critical value) × (standard Error of the mean)
The critical value is obtained using the confidence level, but the margin of error cannot be obtained from just that, we still need the standard error of the mean.
Hence, this statement is false.
f) The confidence level can be determined if you know only the margin of error.
This is another way of phrasing statement E. Hence, this statement too is false.
g) The margin of error can be determined if you know only the confidence level, population standard deviation, and sample size.
This is true as these is all that is required to obtain the margin of error.
h) The confidence level can be determined if you know only the margin of error, population standard deviation, and sample size.
This can be a bit stressful to obtain, but it is true. It is easy to see this from all the explanation from the beginning to this point.
Hope this Helps!!!
