What is a CI?

  • Using a Statistic to estimate a Parameter

  • It is NOT a probability

  • It is an interval that will cover the true parameter X% of the time

    A confidence interval indicates that 19 out of 20 samples (95%) from
the same population will produce confidence Intervals that contain the
population parameter

  • So we can interpret a CI as

    • "We are X% confident that the true population parameter lies within A and B"

General Math Behind a CI

  • Formula

    • Point Estimate ± Margin of Error

    • Point Estimate ± Critical Value * Standard Error

    Population Parameter mean proportion p difference of means
difference of proportions Sample Estimate x p x Conditions for Use
Simple Random Sample Simple Random Sample np25,n I—p 25 Independent
Simple Random Samples Independent Simple Random Samples n, p 125, m
I—p 25, m pa 25,m I—p 25 Formula n pi-ze x n p, I—PI n,

  • Point Estimate


  • Critical Value

    Confdence level 900/0 950/0 9996 99,90/0 Z value 1.65 1.96 2,58

    1025 lower 0.95 1025 upper

  • Standard Error



  • Confidence Interval

    For Population mean (V) Population mean (V) Population proportion
(p) Sample Statistic x x Margin of Error n

Steps to Calculating a CI

  • Read the problem and outline the STASTICS

  • Check your CONDITIONS

    • Random

    • Independent: N≥10n

    • Normal: n>30


    • Point Estimate ± Critical Value * Standard Error

  • Some Can Calculate Intervals

Practice Questions

  1. The effect of drugs and alcohol on the nervous system have been the subject of considerable research. Suppose a neurologist is testing the effect of a drug on response time by injecting 50 rats with a dose, subjecting each to a stimulus, and recording the response time. The average response time for the 50 injected rats was 1.26s. Assuming the mean response time for a rat that has not been injected with the drug is 1.4s with standard deviation of 0.45, construct a 90% confidence interval to determine if the drug has an effect on response time.

    • Statistics

      • Mean = 1.26

      • Population SD = 0.45

      • n = 50

      • CL = 90%

    • Conditions

      • Random: Assume rats are a random sample

      • Independent: N > 10n

      • Normal: n > 50

    • Calculate



    • Calculate by calculator

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2-PropZTest... Z I nt.erval... : T Interval... 9 : Z-SampZ1nt...

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    T 84 u E t 1

    • Calculated using Z test
  • Interpret

    • We are 90% confident that the true mean response time for rats given the new drug is between 1.16s and 1.36s.

    • 1.4s is not in the interval, so we have evidence the new drug make rats faster

  1. There are two fire stations in a town, one in the northern half and one the southern half. The one in the northern part is known to respond to calls within 4 min. The council members in the town are worried that the southern fire station isn’t as good so they hire a statistician. The statistician collects a random sample of 50 call/responses from the southern fire station. The mean response time is 5.3 min with a standard deviation of 3.1. Construct a 95% confidence interval to determine if the council members have cause to worry about the southern station

    • Statistics

      • Mean = 3.5

      • Sample SD = 3.1

      • n = 50

      • CL = 95%

    • Conditions

      • Random: Yes

      • Independent: N > 10n = 500

      • Normal: n > 30

    • Calculate


    i f s• (.(r"؟) 3 S (42.) S

    • Calculate by calculator

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2-FrüFZTest.... 7: ZlnEervaI... g: ırth... Bi2-EampT1nL..

    Tl-84 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n

    ![ТИ-84 Pius Stver Editj(n ф TEns [NSTRUMENTS STA1 гоямдтп сам: F4 ](./media/image201.png)

    • Interoperate

      • We are 95% confident that the true population mean response time for the southern fire station is between 4.42 and 6.18 mins

      • 4 is not in the interval, so we do have reason to be concerned.

  1. The US Department of Transportation reported that 75% of all fatally injured automobile drivers were intoxicated. A random sample of 32 records in Carson County, Colorado, showed that 16 involved a drunk driver. Use a 99% confidence interval to determine whether or not there is evidence that indicates the population proportion of driver fatalities related to alcohol is different than 75%

    • Statistics

      • P hat = x / n = 16/32 = 0.5

      • n = 32

      • CL = 99%

    • Conditions

      • Random: Yes

      • Independent: N > 10n = 320

      • Normal: n * p hat > 10 and n * (1-p hat) >10

    • Calculate



    • Calculate by calculator

    Tl-B4 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n

    Tl-B4 Pius Silver Ecfitjm TEXAS INSTRUMENTS FORMAT n


    • Interperate

      • We are 99% confident that the true population proportion of driver fatalties in Carson County is between 27.2% and 72.8%

      • 75% is not in our interval, so it appears that Carson County is lower in the US

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