WACE Methods normal distribution and confidence intervals

A normal variable is described by its mean and standard deviation; standardise with z = (x - mu)/sigma to compare. The sample proportion has mean p and standard deviation sqrt(p(1 - p)/n), and an approximate confidence interval for p is p-hat plus or minus z times sqrt(p-hat(1 - p-hat)/n).

Section One	Calculator-free, 35%, 50 minutes working
Section Two	Calculator-assumed, 65%, 100 minutes working
Questions	5 to 10 in Section One, 8 to 13 in Section Two
Section Two allows	Up to three calculators (CAS assumed), two A4 sheets of notes, a formula sheet

Normal probabilities

About 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
Sketch the curve and shade the region before you find a probability; for a quantile, work backwards from the area.

Confidence intervals

For 95% confidence use z = 1.96; for 90%, z = 1.645; for 99%, z = 2.576.
The margin of error is z times sqrt(p-hat(1 - p-hat)/n): a higher level of confidence widens it, and a larger sample narrows it.
Most, but not all, intervals built this way contain p: a single interval either contains it or does not.

Try 2 questions

Question 1Practice

The random variable

X

is normally distributed with mean

\mu

and standard deviation

\sigma

. Which statement about its probability density function is true?

Question 2Practice

A table of eight approximate 95% confidence intervals for p, one from each group's sample: group 1: (0.174, 0.346); group 2: (0.238, 0.422); group 3: (0.156, 0.324); group 4: (0.304, 0.496); group 5: (0.219, 0.401); group 6: (0.257, 0.443); group 7: (0.192, 0.368); group 8: (0.165, 0.335).
Group	Lower limit	Upper limit
1	0.174	0.346
2	0.238	0.422
3	0.156	0.324
4	0.304	0.496
5	0.219	0.401
6	0.257	0.443
7	0.192	0.368
8	0.165	0.335

Eight groups each took a random sample of 100 from a population and calculated an approximate 95% confidence interval for the proportion

p

. The table shows the intervals (group 1: (0.174, 0.346); group 2: (0.238, 0.422); group 3: (0.156, 0.324); group 4: (0.304, 0.496); group 5: (0.219, 0.401); group 6: (0.257, 0.443); group 7: (0.192, 0.368); group 8: (0.165, 0.335)). The teacher then reveals that

p

= 0.3

. Which statement is correct?

Next step

Take the free test Try the lesson on confidence intervals

Questions people ask

What does a 95% confidence interval mean?

If many samples of the same size were taken and an interval built from each, about 95% of those intervals would contain the true proportion.

How do I make the margin of error smaller?

Take a larger sample or accept a lower level of confidence; quadrupling the sample size halves the margin of error.