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Estimating the Population Mean

Estimating the Population Mean. Assumptions The sample is a simple random sample The value of the population standard deviation (σ) is known Either the population is normally distributed or n > 30 The sample mean is the best point estimate for the population mean.

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Estimating the Population Mean

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  1. Estimating the Population Mean Assumptions • The sample is a simple random sample • The value of the population standard deviation (σ) is known • Either the population is normally distributed or n > 30 The sample mean is the best point estimate for the population mean.

  2. Confidence Interval estimate of the Population Mean μ (with σ known)

  3. Example Suppose an insurance company studies repair costs after rear collisions, and finds the mean repair cost to be $2300 based on a sample of 40 accidents. Suppose the standard deviation is $1025. Find the 95% Confidence Interval

  4. Example So our confidence interval is: $1982.4 < μ < $2617.7

  5. Example $1982.4 < μ < $2617.7 We are 95% confident that the population mean repair cost is contained in this interval.

  6. Problem Usually σ is not known Solution: Use s (the sample standard deviation) to approximate σ. Since this is not as accurate, we can no longer use our good friend the normal distribution. Now we’re going to need the Student’s t-distribution

  7. Estimating the Population Mean(σ unknown) Assumptions: • The sample is a simple random sample • Either the sample is from a normally distributed population or n > 30 The sample mean is the best point estimate for the population mean.

  8. CI for the Population Mean(σ unknown) Where tα/2 has n-1 degrees of freedom Looking up critical t-values from the table Since our confidence level is centered in the distribution, α is the area in the two tails (far right and far left sides). So in the table: Find the column listing for α in the Area in Two Tails Find the row (degrees of freedom) that is the closest to n-1

  9. Example Suppose an insurance company studies repair costs after rear collisions, and finds from a sample of 28 accidents the mean repair cost to be $2300 and the standard deviation to be $1025. Find the 95% Confidence Interval

  10. Example 27 degrees of freedom, 0.05 area in two tails: tα/2= 2.052 So the 95% confidence interval is:

  11. Estimating Sample Size For sigma not known: • Estimate using the range rule of thumb • Do pilot study • Estimate value using previous study

  12. Example Suppose you were testing whether the mean GPA of a group of students is greater than 2.0. What should your sample size be? Depends on what we want the margin of error to be. Suppose we wanted to be 95% confident that the sample mean is within 0.1 of the population mean.

  13. Example Suppose we wanted to be 95% confident that the sample mean is within 0.1 of the population mean. σ unknown, estimate: We would need a sample size of 384

  14. Homework 6.3: 9, (11), 13, 17, 19, (21) 6.4: 1, 5, 9, 13, (19)

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