Transcript 10.1.2 - GEOCITIES.ws

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 68-9599.7 rule)
California Standard 17.0
Students 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
90%
95%
99%
Z*
1.645
1.960
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  z*

n
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
x
.8403  .8368  .8447
 .8404
3
1.
Find the sample mean
2.
Find the standard deviation
of sample means
.0068
x 

 .0039
n
3
3.
Use z* = 2.576 in the formula
to form the confidence
interval
x  z*
4.
Conclusion: I am 99%
confident that the true mean
weight is between 0.8303g
and 0.8505 g


 .8404  2.576  .0039
n
 .8404  .0101
 (.8303, .8505)
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%
x
.8403  .8368  .8447
 .8404
3
1.
Find the sample mean
2.
Find the standard deviation
of sample means
.0068
x 

 .0039
n
3
3.
Use z* = 1.960 in the formula
to form the confidence
interval
x  z*
4.
Conclusion: I am 95%
confident that the true mean
weight is between 0.8328g
and 0.8480 g


 .8404  1.960  .0039
n
 .8404  .0076
 (.8328, .8480)
Step-By-Step Answer: 90%
x
.8403  .8368  .8447
 .8404
3
1.
Find the sample mean
2.
Find the standard deviation
of sample means
.0068
x 

 .0039
n
3
3.
Use z* = 1.645 in the formula
to form the confidence
interval
x  z*
4.
Conclusion: I am 90%
confident that the true mean
weight is between 0.8340g
and 0.8468 g


 .8404  1.645  .0039
n
 .8404  .0064
 (.8340, .8468)
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 68-9599.7 rule)
California Standard 17.0
Students 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