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![Page 1: Confidence Intervals For a Sample Mean. Point Estimate singleUse a single statistic based on sample data to estimate a population parameter Simplest approach.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649f1f5503460f94c37894/html5/thumbnails/1.jpg)
Confidence Intervals
For a Sample Mean
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Point Estimate
• Use a singlesingle statistic based on sample data to estimate a population parameter
• Simplest approach
• But not always very precise due to variationvariation in the sampling distribution
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Confidence intervalsConfidence intervals
• Are used to estimate the unknown population mean
• Formula:
estimate + margin of error
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Margin of errorMargin of error
• Shows how accurate we believe our estimate is
• The smaller the margin of error, the more precisemore precise our estimate of the true parameter
• Formula:
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Confidence levelConfidence level
• Is the success rate of the method used to construct the interval
• Using this method, ____% of the time the intervals constructed will contain the true population parameter
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What does it mean to be 95% What does it mean to be 95% confident?confident?
• 95% chance that is contained in the confidence interval
• The probability that the interval contains is 95%
• The method used to construct the interval will produce intervals that contain μ 95% of the time.
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• Found from the confidence level• The upper z-score with probability p lying to
its right under the standard normal curve
Confidence level tail area z*
.05 1.645
.025 1.96
.005 2.576
Critical value (z*)Critical value (z*)
.05
z*=1.645
.025
z*=1.96
.005
z*=2.57690%95%99%
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Confidence interval for a Confidence interval for a population mean:population mean:
n
zx
*
estimate
Critical value
Standard deviation of the statistic
Margin of error
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Steps for doing a confidence Steps for doing a confidence interval:interval:1) Assumptions –
• SRS from population• Sampling distribution is normal (or approximately
normal)• Given (normal)• Large sample size (approximately normal)• Graph data (approximately normal)
• σ is known
2) Calculate the interval
3) Write a statement about the interval in the context of the problem.
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Statement: Statement: (memorize!!)(memorize!!)
We are ________% confident that the true mean context lies within the interval ______ and ______.
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Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)σ known
We are 90% confident that the true mean potassium level is between 3.01 and 3.39.
A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with σ = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?
3899.3,0101.33
2.645.12.3
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Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)σ known
We are 95% confident that the true mean potassium level is between 2.97 and 3.43.
95% confidence interval?
4263.3,9737.23
2.96.12.3
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99% confidence interval?
Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)σ known
We are 99% confident that the true mean potassium level is between 2.90 and 3.50.
4974.3,9026.23
2.576.22.3
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What happens to the interval as the What happens to the interval as the confidence level increases?confidence level increases?
the interval gets wider as the confidence level increases
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How can you make the margin of How can you make the margin of error smaller?error smaller?• z* smaller
(lower confidence level)
• σ smaller(less variation in the population)
• n larger(to cut the margin of error in half, n
must be 4 times as big)
Really cannot change!
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A random sample of 50 WHS students was taken and their mean SAT score was 1250. (Assume σ = 105) What is a 95% confidence interval for the mean SAT scores of WHS students?
We are 95% confident that the true mean SAT score for WHS students is between 1220.9 and 1279.1
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Suppose that we have this random sample of SAT scores:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for the true mean SAT score? (Assume σ = 105)
We are 95% confident that the true mean SAT score for WHS students is between 1115.1 and 1270.6.
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Find a sample size:Find a sample size:
• If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:
Always round up to the nearest person!
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The heights of WHS male students is normally distributed with σ = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?
n = 43
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In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.
Can you find a z-interval for this Can you find a z-interval for this problem? Why or why not?problem? Why or why not?
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StudentStudent’’s t- distributions t- distribution
• Developed by William Gosset
• Continuous distribution
• Unimodal, symmetrical, bell-shaped density curve
• Above the horizontal axis
• Area under the curve equals 1
• Based on degrees of freedom
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Graph examples of t- curves vs normal curveGraph examples of t- curves vs normal curve
Normal:
T-Curve: 2 dfs
T-Curve: 5 dfs
T-Curve: 30 dfs
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How does How does tt compare to compare to normal?normal?• Shorter & more spread out
• More area under the tails
• As n increases, t-distributions become more like a standard normal distribution
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How to find How to find tt**
• Use Table B for t distributions• Look up confidence level at bottom &
df on the sides• df = n – 1
Find these t*90% confidence when n = 595% confidence when n = 15
t* =2.132
t* =2.145
Can also use invT on the calculator!
Need upper t* value with 5% is above – so 95% is below
invT(p,df)
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Formula:Formula:
n
stx * :Interval Confidence
estimate
Critical value
Standard deviation of statistic
Margin of errorMargin of error
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Assumptions for Assumptions for tt-interval-interval
• Have an SRS from population
• σ unknown
• Normal distribution– Given– Large sample size– Check graph of data
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For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.
Assumptions:
• Have an SRS of healthy, white males
• Systolic blood pressure is normally distributed (given).
• σ is unknown
We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.
)58.118,22.111(273.9
056.29.114
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RobustRobust
• An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated.
• t-procedures can be used with some skewness, as long as there are no outliers.
• Larger n can have more skewness.
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Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.
(70.883, 74.497)
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Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain.
The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.
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Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:
160 200 220 230 120 180 140
130 170 190 80 120 100 170
Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.
(126.16, 189.56)
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A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?
Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories.
Note: confidence intervals tell us if something is NOT EQUALNOT EQUAL
– never less or greater than!
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Some Cautions:Some Cautions:
• The data MUST be a SRS from the population
• The formula is not correct for more complex sampling designs, i.e., stratified, etc.
• No way to correct for bias in data
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Cautions continued:Cautions continued:
• Outliers can have a large effect on confidence interval
• Must know σ to do a z-interval – which is unrealistic in practice