Confidence IntervalsConfidence IntervalsTarget Goal: Target Goal: I can use normal calculations to construct I can use normal calculations to construct confidence intervalsconfidence intervals. . I can interpret a confidence interval in I can interpret a confidence interval in context.context.

8.1b

h.w: pg 482: 13, 17, 19 – 24, 27, 31, 33

Rate your confidenceRate your confidence0 – 100%0 – 100%

How confident (%) are you that you can ...

Guess my weight within 10 pounds?

. . . within 5 pounds?

. . . within 1 pounds?

What does it mean to be within 10 pounds?

What happened to your level of confidence as the interval became

smaller?

How do we construct How do we construct confidence intervals?confidence intervals?

• If the sampling model of the sample mean is approximately normal, we can use normal calculations to construct confidence intervals.

x

• For a 95% confidence interval, we want the interval corresponding to the middle 95% of the normal curve.

• For a 90% confidence interval, we want the interval corresponding to the middle 90% of the normal curve.

• And so on…

• Suppose we want to find a 90% confidence interval for a standard normal curve. If the middle 90% lies within our interval, then the remaining 10% lies outside our interval. Because the curve is symmetric, there is 5% below the interval and 5% above the interval.

Critical Value Critical Value

• Find the z-values with area 5% below and 5% above.

• These z-values are denoted because they come from the standard normal curve, they are centered at mean 0.

• We use to remind you this is a critical value not a standardized z-score that has been calculated form data!

z

*z

*

If we are using the standard normal curve, we If we are using the standard normal curve, we want to find the interval using want to find the interval using zz-values-values..

Confidence level Tail Area invnorm (1-tail area)

90% 95%

99%

*z

90%.05 .05

.05 1.645 invnorm (.95)

Find rest.

.025 invnorm (.975)1.96

.005 invnorm (.995)2.576

95% .025.025 99%.005.005

1.645-1.645 1.96-1.96 2.576-2.576

Upper p Critical ValueUpper p Critical Value • is called the upper p critical value, with

probability p lying to its right under the standard normal curve.

• To find p, we find the complement of C and divide it in half, or find .

• Remember that z-values tell us how many standard deviations we are above or below the mean.

*z

1

2

C

We want to “catch” the central probability C under We want to “catch” the central probability C under the normal curve.the normal curve.

• To construct a 95% confidence interval, we want to find the values 1.96 standard deviation below the mean and 1.96 standard deviations above the mean, or .

• Using our sample data, this is

assuming the population is at least 10n

(10% condition).

1.96

1.96xn

Constructing a Level C Constructing a Level C Confidence IntervalConfidence Interval

• Using our sample data, we want to find

C I =

• The point estimate for μ is .• The margin of error is

• Note that the margin of error is a positive number, it is not an interval.

x zn

xm z

n

Constructing a Level C Confidence Constructing a Level C Confidence IntervalInterval

• State: Identify the population of interest and the parameter you want to draw a conclusion about at what level.

• Plan: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

• Do: If the conditions are met, carry out the inference procedure.

C I =

CI = est + - margin of error

• Conclude: Interpret the results in the context of the problem

x zn

Ex: Video Screen TensionEx: Video Screen Tension Assume σ = 43 mV. Construct a 90 % confidence interval for the mean tension μ of all the screens produced on this day.

• Refer to data in notes.State : Identify the population of interest

and the parameter you want to draw a conclusion about at what level.

• The population is all of the video terminals produced on this given day.

• We want to estimate parameter μ, the mean tension for all of these screens at a 90% level of confidence.

Plan: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

• Since we know standard deviation, we should use a one sample z confidence interval for the population to estimate μ.

Ck: 1) SRS 2) approx. normal ?Graph data: STATPLOT: normal

probability plot :enter data and use last graph choice.

• The normal probability plot is roughly linear so we have no reason to doubt the normality of the population from which the data came.

Do:Do: If the conditions are met, If the conditions are met, carry out carry out the inference procedure.the inference procedure.

n = 20,• Find for 90% confidence level: • Draw picture:

• C I =• C I =• = (290.5, 322.1)

x 306.3 *z * 1.645z

*x zn

43306.3 1.645

2306. 5

03 1 .8

Conclude:Conclude: Interpret the resultsInterpret the results in in the context of the problem.the context of the problem.

• We are 90% confident that the true mean tension in the entire batch of video terminals produced that day is between 290.5 and 322.1 mV.

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Calculating a Confidence Interval

Properties of Confidence Intervals: The “margin of error” is the (critical value) • (standard dev. of statistic) The user chooses the confidence level, and the margin of error follows from this choice. The critical value depends on the confidence level and the sampling distribution of the statistic.

Greater confidence requires a larger critical value The standard deviation of the statistic depends on the sample size n

The margin of error gets smaller when: The confidence level decreases The sample size n increases

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