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8.1 Estimating µ when σ is K nown

8.1 Estimating µ when σ is K nown. Assumptions (about x ):. We have a simple random sample of size n drawn from a population of x values σ (population standard deviation) is known If x distribution is normal, we may have any sample size

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8.1 Estimating µ when σ is K nown

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  1. 8.1 Estimating µ when σ is Known

  2. Assumptions (about x): • We have a simple random sample of size n drawn from a population of x values • σ (population standard deviation) is known • If x distribution is normal, we may have any sample size • If x distribution is unknown, we need a sample size of n 30 (if x distribution is skewed or not mound-shaped, we may need a sample size of 50, or 100, or more)

  3. Point estimate – an estimate of a population parameter given by a single number. • is the pointestimate of • Margin of error – the magnitude of the difference between the sample point estimate and the true population parameter value. • When using as a point estimate for , the marginoferror is the magnitude of or . • We will use probability to give us an idea of the size of the margin of error when we use as a point estimate of

  4. Confidence level and Critical Value • Confidence level – the probability that the value of a parameter falls within a specified range of values • The confidence level may be any value between 0 and 1, but generally, it is a number such as 0.95, 0.98, or 0.99. • For a confidence level c, the critical value is the number (z score) such that the area under the standard normal curve between and equals c. • Ex: Find a number (the critical value) such that 99% of the area under the standard normal curve lies between and . • Use Table 3 = 2.58, therefore 0.99 • 2.58 is the critical value,

  5. Maximal margin of error, E and Confidence Interval • In estimating, we need some kind of measure of how “good” our estimate is. Probability can give us an idea of the size of the margin of error caused by using the sample mean. • Maximal margin of error • A confidence interval for is an interval computed from sample data in such a way that is the probability of generating an interval containing the actual value of .

  6. How to find a Confidence Interval for when is known: • Let be a random variable • Obtain a simple random sample (of size ) of values from which you computed the sample mean • The value of is already known • If you can assume has a normal distribution, any sample size will work; if not, use a sample size of Confidence interval for when is known: where = sample mean of a simple random sample confidence level (0<c<1) critical value for confidence level based on the standard normal distribution (Table 3b) Example 2 p.320 and Guided Exercise 1 p.321

  7. Sample size for estimating the mean How to find the sample size for estimating when is known: • assuming the distribution of sample mean is approximately normal, then where specified maximal error of estimate population standard deviation critical value from the normal distribution for desired confidence level If is not a whole number, increase it to the next higher whole number ** is the minimal sample size for a specified confidence level and maximal error of estimate • Example 3 p.323

  8. 8.1 PROBLEMS: p. 324-327 #1-3,5,8

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