Probability

1. Probability • CONFIDENCE INTERVALS

2. what is a confidence interval? • keeping it simple, for any random variable (X) a confidence interval is the range of outcome values (x) that we can be confident will contain a specified percentage of all results from an experiment. let’s look at it in greater detail...

3. consider a Normal frequency distribution; • the distribution is symmetrical about the population mean μ, the zero-point on the x-axis; • the 68-95-98% populations divisions are central to confidence intervals. Consider the first σ range, from -1σ to +1σ either side of μ. Ideally, we can be confident that 68% of the population lie within 1σ of μ. That is to say, the 68% confidence interval is -1 < μ <+1

4. -1 α 2 Z this term indicates that the given confidence interval (α%) must be halved, as μ is the midpoint. new notation; this is the value of the inverse normal proportion. traditionally, you would find this value by referring to the Inverse-Norm table, which is still provided in examinations. It is expected that you now have the use of a graphic calculator, and that you know how to use it...

5. Casio fx9750G Plus • Mode = STATS • F4 (INTR) • F1 (Z) • F1 (1-S) • F2 (var) - IMPORTANT • enter your data, and press EXE. an example...

6. enter the data • Execute • report the solution

7. Reporting • It is important to always report a Confidence Interval as the calculated limits bracketing the mean. • the NZQA standard is included in the examination resource sheet: NOTE: The correct interpretation of a confidence interval is that over a long-run experiment, x% of the means will lie within the stated limits.

8. -1 α 2 Z σ √n e = x Sample Size • The Confidence Interval is analogous to the margin of error, which is the remainder of the sample. • For example, if we are dealing with a 98% CI, then • error = ±2%, ie the “lack of confidence” • In terms of accuracy, • CI ≡ sample is accurate to within α%, • so, where e = degree of accuracy If we know the degree of accuracy e, we can calculate n, the minimum sample size

9. -1 α 2 -1 α 2 Z σ √n Z x <4 how to calculate sample size • For this, you have to use the inverse norm table. A calculator will not do it for you. For example: • Assume symmetry; μ is mid-point of 68 and 76, ie μ=72, so e=4 • α = 0.96, so α/2 = 0.48, • Gives 2.05 x 6/√n < 4, so square both sides • (2.05 x 6)2/n < 16, so • n = 205.9225/16 = 12.87, rounded = 13 = 2.05

10. summary points • learn to use the Normal Distribution tables; although most problems can be solve solely with a calculator, sample size calculations require the use of the table. • Always express Confidence Intervals in the correct format;