19  BE2 - Confidence Intervals

20 Lecture

There is no lecture for this topic, but you can re-watch part of the lecture on the sampling distribution to refresh your knowledge about confidence intervals!

21 Formative Test

A formative test helps you assess your progress in the course, and helps you address any blind spots in your understanding of the material. If you get a question wrong, you will receive a hint on how to improve your understanding of the material.

Complete the formative test ideally after you’ve seen the lecture, but before the lecture meeting in which we can discuss any topics that need more attention

Question 1

What does a confidence interval represent in statistics?

The average value of the sample data.The exact value of the population parameter.A range of values within which the true population parameter is likely to fall.The highest value in the sample data.

Question 2

How is the confidence level of a confidence interval chosen?

By using standard values such as 95% or 99%.By the range of values in the sample data.The researcher determines the desired level of confidence in capturing the true parameter.By the size of the sample data.

Question 3

What does a wider confidence interval indicate?

Higher confidence in the accuracy of the estimate.A narrower range of values in the sample data.Smaller sample size.Greater uncertainty or less precision in estimating the population parameter.

Question 4

In general, as the sample size increases, what happens to the width of the confidence interval?

It increases.It decreases.It remains unchanged.It becomes equal to the population parameter.

Question 5

What is the relationship between the width of a confidence interval and the level of confidence?

No relationship: Width and confidence level are independent.Direct relationship: Higher confidence leads to a narrower interval.Inverse relationship: Higher confidence leads to a wider interval.Width and confidence level are equal.

Question 6

What is the purpose of a confidence interval in hypothesis testing?

One can reject a null hypothesis whose hypothesized value lies outside of a X% confidence interval, with a p-value of 1-(X/100).One can reject a null hypothesis whose hypothesized value lies outside of a X% confidence interval, with a p-value smaller than = 1-(X/100).One can reject a null hypothesis whose hypothesized value lies outside of a X% confidence interval, with a p-value smaller than = X/100.One can reject an alternative hypothesis whose hypothesized value lies outside of a X% confidence interval, with a p-value smaller than = 1-(X/100).

Question 7

A researcher collected data from a sample of 80 participants. The sample mean is 72, and the standard deviation is 2.5. What is the 95% confidence interval?

[69.50, 74.50][72.45, 72.55][67.10, 76.90][71.72, 72.28]

Question 8

What is the correct statement about a 95% confidence interval?

There is a 95% probability that the population value is larger than the lower bound and smaller than the upper bound of the confidence interval.The 95% confidence interval is a procedure with 95% probability of providing an interval that contains the population value.The 95% confidence interval contains the population value with 95% probability.95% of all confidence intervals contain the population value.

Question 1

A confidence interval represents a range of values within which the true population parameter is likely to fall.

Question 2

The confidence level of a confidence interval is chosen by the researcher to determine the desired level of confidence in capturing the true parameter.

Question 3

A wider confidence interval indicates greater uncertainty or less precision in estimating the population parameter.

Question 4

As the sample size increases, the width of the confidence interval decreases, indicating increased precision in estimating the parameter.

Question 5

The relationship between the width of a confidence interval and the level of confidence is inverse: higher confidence leads to a wider interval.

Question 6

The purpose of a confidence interval in hypothesis testing is to assess the range of values where the population parameter might lie and make decisions about hypotheses.

Question 7

The standard error of the sample mean is calculated by dividing the standard deviation (2.5) by the square root of the sample size; add +/- 1.96* the standard error to the mean.

Question 8

The probability is in the procedure of calculating a confidence interval. We cannot make any probability statements about the population value, or about the results of specific confidence intervals.

22 Tutorial

22.1 Confidence Intervals

In this assignment we will work on some questions regarding the confidence interval. We focus on the confidence interval around the mean, but everything you learn can also be applied to confidence intervals around the mean difference or regression coefficients.

Finish the following sentence. The confidence interval around the mean is constructed around the Sample mean Population mean (under H0)

In the population the variable IQ is normally distributed with \(IQ \sim N(\mu=100, \sigma=15\).

Imagine that we drew 2000 samples from the population. For each of the samples we would calculate a 90% confidence interval around the sample mean. If you had to make a guess, how many intervals would you expect to contain the value 100?

Imagine we drew a sample from the population and we calculated the 95% confidence interval around the sample mean for a particular variable. The lower bound of the confidence interval is equal to 85 and the upper bound to 95.

Which of the following statements is correct?

The probability that the population mean lies between 85 and 95 is 95%.There is a 95% probability that if we would draw a new sample for the same population, the true population value lies between 85 and 95.There is a 95% probability that the population mean lies between 85 and 95.There is a 95% probability that a confidence interval calculated based on a sample from this population contains the true population value.

Confidence intervals are interpreted in terms of long-run probability. IF we could draw a huge number of samples from the population, 95% of those samples would provide a confidence interval that contains the population mean.

We can never know whether one specific confidence interval contains the population value, however.

So we can NEVER draw a conclusion like “there is a 95% probability that the population mean lies between 85 and 95”.

Recall the first lecture, in which I explained the idea of a “random experiment”. Think of a 95% confidence interval as a random experiment with a 95% probability of containing the population value. One specific confidence interval is not a random experiment. Whether the population mean lies within the interval is not a matter of probability. It either does or it does not. We just don’t know which of these is true.

Imagine a population with variable X, where \(X \sim N(\mu 50, \sigma = 10)\)

Assume a confidence level of 95% for all intervals.

You plan to draw a sample with \(n=20\) and compute a 95% confidence interval. What’s the probability that this interval will contain 50? %

Your colleague has already drawn a sample of \(n=20\). What’s the probability that their confidence interval includes 50? Can’t say 95% 100% 0%

If you would draw 20 samples, how many samples would you expect the confidence interval to contain the value 50?

True or false: if you draw 100 samples, 95 of them will provide a confidence interval that contains the population value. TRUE FALSE

This is false because the phrase “will provide” is not a probability statements, but a deterministic one.

“The number of 95% confidence intervals out of 100 samples that contain the population value” is a random experiment. We expect an outcome of 95, but if we conduct this random experiment, the observed outcome may differ a little, e.g. 93, 94, 97 times are all fine.

All else being equal, what would you expect to happen to the confidence intervals of smaller samples? They become wider They stay the same They become narrower

By increasing the sample size, our estimate becomes more precise. This will lead to more narrow confidence intervals.

Mathematically it also makes sense, because the confidence interval is based on the standard error. Remember that the formula for the standard error is $SE = .A smaller sample size leads to a smaller standard error, which leads to a narrower interval.

Note that this does affect the probability of confidence intervals containing .

If the standard deviation increases (and everything else stays the same) the confidence interval will become narrower stay the same become wider.

If we change the confidence level to 90%, the interval will stay the same become narrower become wider and more fewer the same number of intervals will contain \(\mu\).