There is a 95% chance that the true

garriganj

garriganj

Answered question

2022-07-05

There is a 95% chance that the true mean is between 1 and 3, if our sample's 95% confidence interval is between 1 and 3.

 

Answer & Explanation

Andre BalkonE

Andre BalkonE

Skilled2023-05-22Added 110 answers

To solve the given problem, let's first understand the concept of a confidence interval.
A confidence interval is a range of values within which we can be reasonably confident that the true population parameter lies. In this case, we are interested in estimating the true mean.
Given that the sample's 95% confidence interval is between 1 and 3, we can interpret this as follows:
Let μ represent the true population mean.
The given confidence interval, [1,3], means that we are 95% confident that the true population mean μ falls within this interval.
Mathematically, this can be expressed as:
P(1μ3)=0.95
To solve for the probability, we need to find the Z-score associated with a 95% confidence level. The Z-score corresponds to the number of standard deviations away from the mean.
The Z-score for a 95% confidence level can be found using a standard normal distribution table or calculator. For a two-tailed test, the Z-score is approximately 1.96.
Now, we can express the confidence interval in terms of Z-scores:
1μ3
Dividing both sides by the standard deviation, σ, gives:
1σμσ3σ
Since we are dealing with a standard normal distribution, the Z-scores are equivalent to the standard deviations:
1.96μσ1.96
Now, we can solve for the standard deviation, σ:
μσ=1.96
Solving for σ:
σ=μ1.96
Therefore, the standard deviation is equal to the true mean divided by 1.96.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in College Statistics

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?