ConfidenceInterval for apopulation mean
Section 10.1
Things to remember:
Proportions:Means:
 np ≥ 10normal if
 n(1 – p) ≥ 10                     population is;
CLT
                Population ≥ 10*n
To check forNormal approx.:
To check forIndependence:
Conditions for constructing aconfidence interval
The construction of a confidence interval :
Data come from an SRS from the population of interest
Observations are independent.
The sampling distribution is approximately normal
Constructing aConfidence Interval
Estimate ± margin of error
Estimate ± (critical value)(std. dev. of statistic)
    (statistic)
𝑥 𝑜𝑟 𝑝
𝜎 𝑥 or 𝜎 𝑝 𝜎 𝑛 𝑜𝑟 𝑝 1−𝑝 𝑛
Critical Value
Depends on:
The confidence level (ex. 95% confident)
The sampling distribution
Critical Value z*
The number z* with probability p lying to itsright under the standard normal curve iscalled the upper p critical value of thestandard normal distribution.
un-10-01.jpg 00017C68 production B8414D3D:
Example 10.4, pg. 544Finding z*
If we want to find the 80% confidenceinterval…
figure-10-05.jpg 00017C68 production B8414D3D:
figure-10-05.jpg 00017C68 production B8414D3D:
What is the area just under A?
.9000
Look this up in Table A!!!!
A
figure-10-05.jpg 00017C68 production B8414D3D:
What is the area just under A?
.9000
Look this up in Table A!!!!
z* = 1.28
A
Common Confidence Intervals
figure-10-06.jpg 00017C68 production B8414D3D:
ConfidenceLevel
Tail Area
z*
90%
0.05
1.645
95%
0.025
1.960
99%
0.005
2.576
To construct aConfidence Interval:
State – know what parameters we’reestimating & at what confidence level
Plan – choose method & check conditions
Do – if conditions are met, performcalculations
Conclude – interpret the interval in thecontext of the problem
Example 10.5, p. 546
A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. To much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mV). Some variation is inherent in the production process. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is 𝜎=43 mV, and suggests the tension readings of screens produced on a single day follow a normal distribution quite closely. Here are the tension readings from an SRS of 20 screens from a single day’s production. Construct a 90% confidence interval for the mean tension µ of all the screens produced on this day.
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. To much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mV). Some variation is inherent in the production process. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is 𝜎=43 mV, and suggests the tension readings of screens produced on a single day follow a normal distribution quite closely. Here are the tension readings from an SRS of 20 screens from a single day’s production. Construct a 90% confidence interval for the mean tension µ of all the screens produced on this day. State: know what parameters we’re estimating & at what confidence level Population of interest: All of the video terminals produced on the day in question. Parameter we’re drawing conclusion about: Want to estimate µ, the mean tension for all of these screens.
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. To much tension will tear the mesh and too little will allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mV). Some variation is inherent in the production process. Careful study has shown that when the process is operating properly, the standard deviation of the tension readings is 𝜎=43 mV, and suggests the tension readings of screens produced on a single day follow a normal distribution quite closely. Here are the tension readings from an SRS of 20 screens from a single day’s production. Construct a 90% confidence interval for the mean tension µ of all the screens produced on this day. Plan – choose method & check conditions Inference procedure: mean Verify conditions: SRS from population of interest? Sampling distribution of 𝑥 approximately normal? Independent?
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
Yes!
Yes!
Yes!
Construct a 90% confidence interval for the mean tension µ of all the screens produced on this day. Do – if conditions are met, perform calculations Find 𝑥 : Use the confidence interval formula 𝑥 ± 𝑧 ∗ 𝜎 𝑛 For 90% confidence level, the critical value is 𝑧 ∗ =1.645 The 90% confidence interval for µ is: 𝑥 ± 𝑧 ∗ 𝜎 𝑛 =
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
𝑥 =306.3 mV
306.3±(1.645) 43 20 =
306.3±15.8=
(290.5, 322.1)
Construct a 90% confidence interval for the mean tension µ of all the screens produced on this day.
Conclude – interpret the interval in the context of theproblem
We are 90% confident that the true mean tension in the entire batch ofvideo terminals produced that day is between 290.5 and 322.1 mV.
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
Confidence Level for aPopulation Mean µ
Choose an SRS of size n from a population having unknown mean µ and known standard deviation 𝜎. A level C confidence interval for is 𝑥 ± 𝑧 ∗ 𝜎 𝑛 Here z* is the value with area C between –z* and z* under the standard normal curve. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.
P. 548:  10.5, 10.7 b
Due:  Wednesday
Homework