Size & Power of Test (Edexcel A-Level Further Mathematics): Revision Notes

Example Take the data set

12, 16, 10, 9, 12

If we are given that these are from a normally distributed population, the $\overline{x}$ is a point estimate for μ and $s^2$ for $\sigma^2$.

However, due to the nature of statistical error, the point estimate may not be completely accurate; however, we can give bounds within which we know the population parameter we are estimating is likely to lie. These are called confidence intervals.

Example A point estimate from a sample of size 50 is calculated to be 26.6 for the mean μ. Given the population is known to have standard deviation σ = 6, construct a 95% confidence interval for μ.

1. Draw a diagram to aid in determining the correct z-value to calculate:

CALCULATE Z-Value associated with 0.975

2. The mean μ lies within the confidence interval to the given degree of confidence according to the following formula:

Where:

• $\overline{x}$ is the centre of the interval (sample mean).
• $Z$ is how many standard deviations we want our boundary to lie from $\overline{x}$.
• $\frac {\sigma}{\sqrt{n}}$ is the standard deviation of the sampling distribution to reach our Z-value to fit our desired distribution.