Estimating the population mean

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Estimating the population mean
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Objective

• Our objective is to predict or estimate the value
  for the unknown mean m based on data from a
  sample.

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Assumptions

• 1. The sample is a simple random sample.
• 2. Either the population is normally distributed
  or n gt 30 or both conditions are true.
• 3. The value of s is known.

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Point Estimate

• The point estimate for m is simply the sample
  mean.
• Example If we study 25 college students and
  find the average number of hours working is 20
  hours, then we can estimate m to be 20 hours.

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Confidence Intervals

• Confidence intervals are a range of values in
  which the population mean should lie.
• Confidence levels gives the success rates of the
  procedure used to construct the interval.
• Typical levels are 90, 95, and 99.

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90 level

• We need to find the z score that has 90 of the
  area from z to z

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95 level

• We need to find the z score that has 95 of the
  area from z to z

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99 level

• We need to find the z score that has 99 of the
  area from z to z

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Error

• The error is the amount that our estimate may
  differ from the sample mean. Round E to match
  the number of decimal places in the sample mean.

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Interval

• The interval is found by

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Example

• A lumber yard checks a random sample from a large
  delivery of plywood boards. The mean thickness
  of 49 boards is 0.980 inches. The population
  standard deviation is 0.056 inches.
• Find the 90 and 95 confidence intervals.

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Important consequence

• By increasing the confidence level, the value of
  E and the size of the interval will be increased.

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• Suppose for the original problem, the sample size
  was 100.

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Important consequence

• By increasing the sample size, the value of E and
  the size of the interval will be decreased.

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Checking on the TI

• Press Stat, Tests

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• Press 7, Z interval.
• Enter the values.

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• Press Calculate
• Note The TI may not give exactly the same
  values.

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Your Turn

• Julia is a jogger who has recorded her times over
  several years for jogging a distance of 2 miles.
  She has found that the standard deviation is 1.80
  minutes. For the last 20 days, her mean time was
  15.60 minutes. Find the 95 confidence interval
  for m, which would be for her times for the last
  year.

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Calculating the sample size needed

• Before conducting a survey, statisticians
  pre-determine the confidence level and amount of
  allowable error.
• Solving the Error formula for n results in

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Important Note

• Since we can not survey a part of a person, the
  calculated value of n must always be rounded up
  to the next integer.

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Example

• A 95 confidence interval for the mean speed of
  cars on a highway is to be constructed which must
  be accurate to within 2 miles per hour. Assume
  the population standard deviation is 9 miles per
  hour. What size sample is required?

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• If the maximum error is 1 mile per hour, what
  size sample is required?

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Important Consequence

• By decreasing the amount of the error, the
  required sample size is increased.

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Your Turn

• An economist wants to estimate the mean income
  for the first year of work for college graduates
  who have had the profound wisdom to take
  Statistics I. How many people must be surveyed
  in order to be 95 confident that the sample mean
  is within 500 of the population mean. Assume a
  previous study has found s 6250.