E STIMATION – C ONFIDENCE I NTERVALS Stats 1 with Liz

AIMS To understand what a confidence interval is To be able to calculate confidence intervals P RIOR K NOWLEDGE Thorough understanding of normal distributions

Using the normal distribution curve below complete the following sentences: About …… of the population is within one standard deviation of the mean About …… of the population is within two standard deviations of the mean About …… of the population is within three standard deviations of the mean

90% C ONFIDENCE I NTERVAL Draw a diagram & label your percentages. We want to find the value of z or –z. You can use whichever option is easier to work with since they’re symmetric. Since our tables show us the probabilities that are less than a given value, I am going to find z when Hence, our z value is This means that our 90% confidence interval for a standard normal distribution is

95% C ONFIDENCE I NTERVAL Draw a diagram & label your percentages. We want to find the value of z or –z. You can use whichever option is easier to work with since they’re symmetric. Since our tables show us the probabilities that are less than a given value, I am going to find z when Hence, our z value is This means that our 95% confidence interval for a standard normal distribution is

99% C ONFIDENCE I NTERVAL – Y OU TRY ! Draw a diagram & label your percentages. We want to find the value of z or –z. You can use whichever option is easier to work with since they’re symmetric. Since our tables show us the probabilities that are less than a given value, I am going to find z when Hence, our z value is This means that our 99% confidence interval for a standard normal distribution is

Y OU ’ VE LEARNED THE THREE MOST COMMONLY ASKED FOR CONFIDENCE INTERVALS. Y OU CAN MEMORISE THESE TO MAKE WORKING SOME QUESTIONS QUICKER. HOWEVER!!! M AKE SURE YOU KNOW HOW TO FIND THEM BY HAND JUST IN CASE THEY ASK YOU FOR A LESS COMMON INTERVAL (92%, ETC )

C ONFIDENCE I NTERVAL F ORMULA In general, the confidence interval for the population mean can be found using the formula: Upper limitLower limit Where is the sample mean is the confidence level you’ve found using your tables is the population standard deviation is the sample size

K EY C ONFIDENCE I NTERVALS FOR ANY NORMAL DISTRIBUTION 90% confidence interval: 95% confidence interval: 99% confidence interval:

E XAMPLE 1 We know that for a 95% c.i., we can use the following formula to find our lower and upper limits: Lower limit: Upper limit: Hence, our 95% c.i. is

E XAMPLE 2 S TATS 1 TEXTBOOK, P G. 134, E X. 7 A machine cuts metal tubing into pieces. It is known that the lengths of the pieces have a normal distribution with a standard deviation of 4 mm. After the machine has undergone a routine overhaul, a random sample of 25 pieces are found to have a mean length of 146 cm. Assuming the overhaul has not affected the s.d. of the tube lengths, determine a 99% C.I. For the population mean length, giving your answer to the nearest centimetre. We know that for a 99% c.i., we can use the following formula to find our lower and upper limits: Lower limit: Upper limit: Hence, our 99% c.i. is to the nearest cm.

W HAT DOES THIS ACTUALLY TELL US ? It is important to understand the meaning of your answer. This particular interval either does or does not include the population mean length – you cannot say which has occurred because it’s not a 100% certainty. Hence, our 99% c.i. is to the nearest cm. O UR SOLUTION WAS : What you can say is that 99% of the intervals constructed in this way will include the population mean length, This is sometimes described as “Being 99% confident that the calculated interval contains

Y OU TRY ! S TATS 1 TEXTBOOK, P G. 141, Q. 8 Pencils produced on a certain machine have lengths, in millimetres, which are normally distributed with a mean of μ and a standard deviation of 3. A random sample of 16 pencils are taken and the length x mm recorded for each one giving ∑x = 2848 (a)State why X bar is normally distributed (b)Construct a 99% confidence interval for the mean a)X has normal distribution b)(176.1,179.9)mm

E XAM Q UESTION June 06 q4

A SSIGNMENTS FULLY read and practise all examples in the Graphing Calculator booklet