Post on 21-Jan-2016
Stats Ch 8
Confidence Intervals for One Population Mean
Estimating a population mean
• Point estimate: Use a statistic to estimate the corresponding parameter. (Often we use a point estimate…that is we use the mean of a sample to estimate the mean of the population, or the st. dev of a sample to estimate the st. dev of a population, etc)
• Most of the time there is sampling error. So we often need confidence intervals
Confidence Intervals
• Confidence-Interval Estimate – Confidence interval (CI): an interval of numbers
obtained from a point estimate (the range: from 100 to 200)
– Confidence level: The confidence we have that the parameters lies with in the CI (the percent: 90% confident)
– Confidence-interval estimate: a combination of the confidence interval and the confidence level (we are 90% confident that mu is between 100 and 200)
Example
• A random sample of the costs of 45 weddings was taken (∑xi = $608,580). The population standard deviation is $5205. Obtain a point estimate for the population mean. Determine a 93% confidence interval for the population mean. Interpret your results.
Ex
• Pg 383 problem 49 (also interpret the results)
CI when sigma is known
• Confidence level vs α: CL = 1 – α…..so using algebra α = 1 – CL
• is the z-score with an area of α to the RIGHT under standard normal curve
z
One-Mean z-Interval Procedure
• Assumptions: Simple random sample, normal distribution or large sample, σ is known
• 1. Sketch picture• 2. Find α and use Table C to find
• 3. Find CI--use and • 4. Interpret the CI
2z
n
zx
2
n
zx
2
CI
• Pg 375 ex 8.3
• How does the range of the CI (precision) effect the CL
Margin of Error
• The margin of error is how far away (above OR below) your CI critical points are from
• Formula:
• How can we decrease the margin of error (which will increase the precision)?
• Pg 379 ex 8.8
x
CI when sigma is NOT known
• Studentized version of which is• There is a different t-distribution for each sample size.
We identify the particular t-distribution by the degrees of freedom which is df = n – 1
• Basic properties of t-curve:– Total area under the curve is 1– Extends indefinitely in both directions as it approaches the
horizontal axis– Symmetric about 0– As df increases, the t-curve looks more like the standard
normal curve
xns
xt
/
The t-table (Table D)
• Uses notation. The t-value with an area of α to the RIGHT
• Pg 388 ex 8.9• Find a CI for one-mean t-interval is almost the exact
same way as with one-mean z-interval• Assumptions: Simple random sample, normal
distribution or large sample, σ is unknown• 1. Sketch picture• 2. Find α and use Table D to find with df = n – 1 • 3. Find CI--use and • 4. Interpret the CI
t
2t
n
stx2
n
stx2
Example t
• Pg 389 ex 8.10
Margin of Error
Pg 393 Ex 8.14
T-distribution examples
• Pg 397 #50, #54