Math 3400 Computer Applications in Statistics Lecture 10 Confidence Intervals

Math 3400 – Confidence Intervals When taking a sample, we estimate how “right” we are in terms of representing the population through the calculation of confidence intervals. Inferential statistics – not descriptive statistics We can estimate the CI for any population parameter – mean, std, proportion.

Math 3400 – Confidence Intervals The mean or average is the most commonly estimated population parameter. Example: A product claim might be: The new Milemaster tire from Firestone will last 50,000 miles. - Question – since descriptive statistics are more accurate, why wouldn’t we just calculate the population parameter for tire life directly?

Math 3400 – Confidence Intervals The formula for calculating a confidence interval is: x + (t*s)/SQRT(n) Where, x is the sample average t is the appropriate t-statistic s is the sample standard deviation n is the sample size

Math 3400 – Confidence Intervals Note regarding the t-distribution… The t-distribution is similar to the z, or normal, distribution. However, it has slightly more space in the tails, accommodating more uncertainty and variability than the z-distribution. It is generally used if σ is unknown, or if n<30.

Math 3400 – Confidence Intervals Example from the book (Table 8.1 on page 329): Brand X: x = s = n = 10 t=2.26 (df = 9) So, (2.26*10.54)/SQRT(10) =

represents a 95% confidence interval. This means that 95/100 times the estimated population mean will fall within the confidence interval. You can think of the 7.5 as the margin of error (E). This is the maximum value would we expect to “off” on our prediction. x – E < μ < x + E Math 3400 – Confidence Intervals

Question – what if we wanted to maintain this interval, but increase our confidence? What would our sample size need to be (currently, n=10).

Math 3400 – Confidence Intervals What do we know? Brand X: x = s = n = ? t=3.29 (df = 9) this is the t-statistic for 99% confidence.

Math 3400 – Confidence Intervals [(3.29*10.54)/SQRT(n)]= *10.54/SQRT(n) = /SQRT(n) = 7.5 SQRT(n) = 4.62 n = What happened to the sample size?

Math 3400 – Confidence Intervals Fun EXCEL Exercise

Math 3400 – Confidence Intervals Firestone wishes to construct a CI on the mean life of its new Milemaster tire. The firm takes a simple random sample of 900 tires and tests them until tread thickness is below federal standards. Here are the descriptive stats: x = 47,500 s = 3000 Determine the 90%, 95% and 99% confidence intervals.

What if Firestone wanted to utilize a 99% CI, but decrease the size of the interval? Math 3400 – Confidence Intervals

There are two specific points worth noting: 1.CIs are only valid for STABLE processes where the average is relevant; 2.As sample size goes up, the width or uncertainty about the population parameter goes down. Math 3400 – Confidence Intervals

If I pull one observation at random from a population, how certain can I be of its value? This question refers to a prediction interval of one observation. The relevant formula is: x+ (t*SQRT(s 2 + (s 2 /n))) This is the uncertainty related to a single observation within a sample This is the uncertainty related to sampling

Math 3400 – Confidence Intervals If we select one observation at random, we can be 95% certain that the value will lie between: *(SQRT( ( /10))) = Why is this interval larger than the first interval calculated?

Math 3400 – Confidence Intervals If we are interested in estimating a population proportion of discrete units – like voters or customers – from a sample, we use a proportion confidence interval. Application examples: What % of voters will cast a vote for Bush? What % of customers will attrite over the next 12 months?

University of Connecticut Poll. Feb , N=1,121 registered voters nationwide. MoE ± "If the 2004 presidential election were being held today, and George W. Bush was running as the Republican candidate and John Kerry was the Democratic candidate, for whom would you vote?". George W. Bush John Kerry Other (vol.) Don't Know %% 2/12-16/ Math 3400 – Confidence Intervals

The formula for calculating a proportion confidence interval is: p + z α/2 *SQRT(((pq)/n)) Where, p is the sample proportion average z is the appropriate z-statistic a is the designated alpha value (1-Confidence) q is 1-p n is the sample size Why is there no standard deviation variable?

Math 3400 – Confidence Intervals Example: 829 Minnesotans were surveyed by the Minneapolis-St. Paul Star Tribune to reveal opinions about the “photo-cop”…51% were opposed to the concept. Find the 95% confidence interval for the general population.

Math 3400 – Confidence Intervals What do we know? p =.51 q =.49 z = 1.96 a =.05/2 n = 829 So the confidence interval here is:

51%+ 3.4% represents a 95% confidence interval. This means that we are 95% confident that this interval contains the actual percentage of people opposed to the photo-cop. Question – what if we wanted to maintain this interval, but increase our confidence? What would our sample size need to be (currently, n=829). Math 3400 – Confidence Intervals