*MATH 225N Confidence Intervals*

MATH 225N Confidence Intervals

I took a data set of n = 35 values for Height measured in Inches and I used our Week 6 Excel spread sheet Calculator ( please see Week 6 Files area after first clicking Files along the left of the computer screen ) and I calculated 4 confidence intervals with the same sample mean and same sample size and same population standard deviation.

The 4 confidence levels were ( respectively ) 99% 95% 90% 85%

Notice that the 4 margins of error were ( respectively ) 1.57 inches 1.19 inches 1.00 inches 0.88 inches

If this is not a coincidence – that is – if this trend / pattern holds up in general ( at least for the confidence intervals for one population mean that we study and learn about in this course ) then how would you put that pattern / trend into words ??

In other words, all other things being equal ( the same or “fixed” ) , what happens to the margin of error when the confidence level is increased ?? ( or decreased ?? )

Thanks Friends and please see attached ( see the tab along the bottom for the z confidence interval – sample size larger than 30 – the left most tab along the bottom of the spread sheet ) 😉

The original height data used are attached to the next Post

That is where the sample mean and population standard deviation came from.

I used population standard deviation rather than sample standard deviation because the sample size of n = 35 was larger than 30

I tried another experiment with confidence intervals.

I calculated 4 95% confidence intervals all using the same sample mean of 65.8857 inches and the same population standard deviation of 3.5998 inches.

The only difference between the 4 95% confidence intervals is that I based the calculations of the confidence interval endpoints ( left endpoint and right endpoint ) on 4 different sample sizes of n = 35 and then n = 70 and then n = 105 and then n = 140 .

The 4 ( respective ) margins of error were 1.19 inches 0.84 inches 0.69 inches 0.60 inches

If this is not a coincidence – if the confidence interval for one population mean that we study and learn about in this course always behaves according to this trend / pattern – how can you put this trend / pattern into words ??

That is, all other things being equal ( the same or “fixed” ) then what happens to the margin of error as the sample size is increased ??

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Another way of thinking about this is – if you use a larger sample size you are incorporating “more information” into

the analysis – so what is the “reward” for doing that in this case ??

The n = 35 spread sheet is attached to this Post and the spread sheets for ( respectively ) n = 70 and then n = 105 and then n = 140 are attached to 3 further Posts just below here.

Thanks Friends and please see attached ( see the tab along the bottom for the z confidence interval – sample size larger than 30 – the left most tab along the bottom of the spread sheet ) 😉

Be sure to focus on just the 95% confidence interval and its margin of error in each of the 4 spread sheets

Thanks Friends and Very Best Wishes and Good Luck during this Week 6 !!

This Week is a busy one because we have to complete the Week 6 Knewton Quiz and also get off to a great start on the Week 7 lab turn in assignment in the SAME Week !!

Good Luck and THANK YOU so much for your hard work !!

Intuition tells me that bigger sampling increases your odd of getting closer to the true value, therefore, error is reduced.

The law of large numbers says as your samples get larger, the mean of the sample gets closer to the true mean and the standard deviation gets smaller (Holmes et al., 2018). I’m not clear yet how that relates to margin of error as I said before, they seem kind of similar to me. Please clarify that some more.

Holmes, A., Illowsky, B., & Dean, S. (2018). *Introductory business statistics*. OpenStax.

I found this you tube video that explains the correlation between standard deviation, margin of error, and CI. Maybe this may help you understand it better. It helped me.

https://youtu.be/9g2MHYYKpNMLinks to an external site.

Barb

References:

Statisticsfun. (2011). How to calculate margin of error and standard deviation. *You tube. *Retrieved from https://youtu.be/9g2MHYYKpNMLinks to an external site.

The margin of error gets larger with the confidence level because if we want to be more confident that the population mean we are looking for is in the interval, we need to make the interval as wide as possible.

I’m still a little confused about the relationship between the margin of error and the standard deviation, though, if there is one.

This gets me too. What I have read is they seem to be the same, but what I took from everything I looked up, and that was multiple sights, is that the standard deviation and the margin of error work together? Not sure if that is right, but it is what I took from my research of the two. Per Glen (2020). “The margin of error is the range of values below and above the sample statistic in a confidence interval. The confidence interval is a way to show what the uncertainty is with a certain statistic (i.e. from a poll or survey). With Standard Deviation, you must know the population parameters to calculate it. The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x Standard deviation* for* the population. Margin of error = Critical value x Standard error of the sample” (p. 1).

### Reference:

Glen, S. (2020). “Margin of Error: Definition, How to Calculate in Easy Steps” From StatisticsHowTo.com.Links to an external site.* Elementary Statistics for the rest of us!* https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/margin-of-error/Links to an external site.

This Week 6 our confidence intervals are

( point estimate – margin of error , point estimate + margin of error )

which by the way means that the entire width / length of the confidence interval is two times the margin of error.

Also the midpoint of the confidence interval is the point estimate here in Week 6 .

Meaning that if we add the confidence interval endpoints together and divide by two, we get the point estimate back.

The point estimate for the confidence interval in the mean context is the sample mean xbar

The point estimate for the confidence interval in the proportion context is the sample proportion phat

It is important to note that within Knewton sometimes they use the symbol p’ for the sample proportion rather than the symbol phat

The formula for the margin of error contains several elements / pieces / ingredients that don’t change no matter what confidence level is used to calculate the confidence interval ( to calculate the confidence interval endpoints ) .

The only piece / element / ingredient in the formula for the margin of error that changes with different confidence levels is the critical value.

In some cases this Week 6 we are talking about a critical z value here and in some cases this Week 6 we are talking about a critical t value here.

But focusing on the cases where we use a critical z value during this Week 6 , the critical z value changes from 1.645 to 1.96 to 2.575 as the confidence level goes from 90% to 95% to 99%

So in calculating the margin of error for the confidence interval for each of these confidence levels, we multiply some fixed quantity times an ever increasing number ( ever increasing critical z value ) as the confidence level goes higher and higher.

That is why “all other things being equal,” the margin of error increases with increasing confidence level.

**All this stuff mentioned here recently would be a really good answer to one of the important questions near the end of the Week 7 lab assignment next Week 7 !!**

Thanks Barb and Elaine and Everyone have a Terrific day and a Fantastic Week 6 !!

🙂