General Confidence Intervals

Section 10.1.2

Starter 10.1.2

• A shipment of engine pistons are supposed to have diameters which vary according to N(4 in, 0.1 in)

• A sample of 10 pistons has an average diameter of 4.05 inches

• State a 95% confidence interval for the true mean diameter of all the pistons

Today’s Objectives

• Find the z* critical value associated with a level C confidence interval

• Find a confidence interval for any specified confidence level C– (In other words, let’s remove the need for the 68-95-

99.7 rule)

California Standard 17.0Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error

Confidence Intervals• A level C confidence interval for a

parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter.

• To get confidence level C we must catch the central probability C under a normal curve

• So we define a value called z* such that the area under a standard normal curve between –z* and +z* is C

Finding the z* values• Suppose we want a 90% confidence interval• Then 90% of the area under the curve must be between

–z* and +z*• Since the curve is symmetric, that means that 5% of the

area is below –z* and 5% is above +z*– So how much area is below +z*?

• Search Table A for the z value that has 95% of the area to its left– z is between 1.64 and 1.65, so we can use 1.645

• Use the calculator to get the same result– invnorm(.95) = 1.645

• Use Table C to get the same result– z* values are in the bottom row above the values for C

The important z* values

• You have found the z* associated with a 90% C.I.• Now find z* for a 95% C.I. and for a 99% C.I.• Summarize your results in a simple table

Confidence

Level

Z*

90% 1.645

95% 1.960

99% 2.576

Using z* to form a C.I.

• The form of confidence intervals is

estimate ± margin of error• The margin of error is a number of standard deviations

– In our example yesterday, we used 2 s.d.• Since z* is measured in standard deviations, multiply by

the s.d. of the sampling distribution to get margin of error• Then add and subtract the margin to the estimate• So here is the formula for forming a level C confidence

interval:*x zn

Example 10.4

• Repeated weighings of the active ingredient in a painkiller are known to vary normally with a standard deviation of .0068g

• Three specimens weigh:

0.8403g 0.8363g 0.8447g

• Form a 99% confidence interval for the mean weight of the ingredient.

Step-By-Step Answer

1. Find the sample mean

2. Find the standard deviation of sample means

3. Use z* = 2.576 in the formula to form the confidence interval

4. Conclusion: I am 99% confident that the true mean weight is between 0.8303g and 0.8505 g

.8403 .8368 .8447.8404

3x

.0068.0039

3x

n

* .8404 2.576 .0039

.8404 .0101

(.8303, .8505)

x zn

Example 10.4 Modified

• Repeated weighings of the active ingredient in a painkiller are known to vary normally with a standard deviation of .0068g

• Three specimens weigh:0.8403g 0.8363g 0.8447g

• Form a 95% and a 90% confidence interval for the mean weight of the ingredient.

Step-By-Step Answer: 95%

1. Find the sample mean

2. Find the standard deviation of sample means

3. Use z* = 1.960 in the formula to form the confidence interval

4. Conclusion: I am 95% confident that the true mean weight is between 0.8328g and 0.8480 g

.8403 .8368 .8447.8404

3x

.0068.0039

3x

n

* .8404 1.960 .0039

.8404 .0076

(.8328, .8480)

x zn

Step-By-Step Answer: 90%

1. Find the sample mean

2. Find the standard deviation of sample means

3. Use z* = 1.645 in the formula to form the confidence interval

4. Conclusion: I am 90% confident that the true mean weight is between 0.8340g and 0.8468 g

.8403 .8368 .8447.8404

3x

.0068.0039

3x

n

* .8404 1.645 .0039

.8404 .0064

(.8340, .8468)

x zn

Conclusion• Describe the change in the confidence

intervals we found as we changed C.

• As C decreased from 99% to 95% to 90% the intervals got narrower.– In other words, more accurate.

• What did we give up to get the increased accuracy?

• We reduced confidence. In the last case, we used a method that gives correct results in 90% of all samples, not 99%.

Today’s Objectives

• Find the z* critical value associated with a level C confidence interval

• Find a confidence interval for any specified confidence level C– (In other words, let’s remove the need for the 68-95-

99.7 rule)

California Standard 17.0Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error

Homework

• Read pages 513 - 518

• Do problems 5, 7, 8