Question 1Practice
The mass of an apple from an Adelaide Hills orchard has mean 150 g and standard deviation 12 g. A bag holds 9 apples chosen at random, with independent masses. What are the mean and standard deviation of the total mass?
SACE Methods normal distribution and confidence intervals
A normal variable is set by its mean and standard deviation; standardise with z = (x - mu)/sigma. The mean of a sample of n has standard deviation sigma/sqrt(n), and a confidence interval is the sample estimate plus or minus z standard errors, for a mean or for a proportion.
| The examination | 130 minutes, 100 marks, 30% of the subject |
|---|---|
| Booklets | Two question booklets of 50 marks, about 65 minutes each |
| Allowed | A graphics calculator throughout, two A4 sheets of handwritten notes, a formula sheet |
| Answers | Working shown; three significant figures unless a question says otherwise |
Normal probabilities and sampling
- About 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
- The sum of n independent observations has mean n mu and standard deviation sigma sqrt(n); their mean has mean mu and standard deviation sigma/sqrt(n). For large n both are close to normal, whatever the shape of X.
Confidence intervals
- For 95% confidence use z = 1.96; for 90%, z = 1.645.
- For a mean: x-bar plus or minus z sigma/sqrt(n). For a proportion: p-hat plus or minus z sqrt(p-hat(1 - p-hat)/n).
- A claimed value inside the interval is consistent with the sample; outside, it is doubtful. Neither proves the claim.
Try 2 questions
Question 2Practice
A random sample of 36 bunches of shiraz grapes from a Barossa Valley vineyard has a mean mass of 152.4 g and a standard deviation of 18 g. Given and , which of the following is a 95% confidence interval for , the mean mass in grams of all the bunches in the vineyard?
Next step
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 value.
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.