Confidence Interval Estimation

Confidence Intervals on and

• An interval estimator is a formula that tells us how to use sample data to calculate an interval that estimates a population parameter.

• The confidence coefficient is the probability that an interval estimator encloses the population parameter.

• The confidence level is the confidence coefficient expressed as a percentage.

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Useful Probabilities for Normal Distributions68%95%99%

Normal Distribution for the Mean

• Confidence intervals assume that the sample means are normally distributed.

Useful Distribution Revisited

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Confidence Intervals

• A 95% confidence interval represents a range of values within which you are 95% certain that the true population mean exists.– One interpretation is that if 100 different samples

were drawn from the same population and 100 intervals were calculated, approximately 95 of them would contain the population mean.

5

Confidence Intervals Sometimes Miss

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z2

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x n

Large-Sample Confidence Interval for

• where is the z value with an area to its right and

• The parameter is the standard deviation of the sampled population and n is the sample size.

• When is unknown (most cases) and n is large, the value of can be approximated by the sample standard deviation, s.

Commonly Used Values of

z2

Confidence Level1001( )

2z2

90% .10 .051.64595% .05.0251.9699% .01.0052.575

Example

• Suppose you sampled 400 college students to determine the average soft drink consumption of college students. The results are a sample mean of 20.1 and a sample standard deviation of 0.5. What is the 95% CI for the average soft drink consumption for college students?

Interpretation of a Confidence Interval for a Population Mean

• We can be that lies between the lower and upper bounds of the confidence interval.

• The statement reflects our confidence in the estimation process rather than in the particular interval that is calculated from sample data.

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Example

• A tire manufacturer is testing a new compound for tread wear. Tires made with the compound are placed on a machine that simulates road wear, and the amount of tread left after the equivalent of 40,000 road miles is recorded. The mean amount of tread left after a test of 10 tires was 3.6 mm. Construct a confidence interval for the population mean amount of tread left after 40,000 miles at the 99% level of confidence. Assume the population distribution is normally distributed with a standard deviation of 0.2 mm.

Answer

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Example

• An automobile dealer plans to order enough cars to have a 90-business-day inventory. Over the past 30 business days the dealer has sold an average of 20 cars a day. Construct a Confidence Interval for the population mean number of cars sold per day at the 95% level of confidence. Assume that the distribution of cars sold per day is approximately normal with a standard deviation of 5 cars.

Answer

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Small-Sample Estimation of a Normal Probability

• The sample standard deviation s may provide a poor approximation of the population standard deviation when the sample size is small.

• In these cases the we use the t-statistic rather than the z-statistic.

t-statistic

• Formula -

• t-statistic is more variable than the z-statistic. The variability depends upon the sample size, n.

• The t-statistic has (n-1) degrees of freedom (df).

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Small-Sample Confidence Interval for

• Formula -

• The confidence interval using the t-statistic is wider than the corresponding confidence interval using the standard z-statistic.

• Curves:

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Example

• A sample of monthly sales for 20 Circle K convenience stores shows mean sales to be $30,000 with a standard deviation of $16,000:– Calculate a 95% confidence interval for

monthly sales using a t-statistic.– Calculate a 95% confidence interval for

monthly sales using a z-statistic.

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Large-Sample Estimation of a Binomial Probability

• Often we want to the know the proportion of a population such as the proportion of smokers, proportion of smokers that prefer a specific brand, proportion of viewers that remember a commercial, etc.

• How do we estimate p, the proportion of successes in the sample?

Estimation of p

• One logical answer is to calculate p as follows:– where – x = number of successes– n = number of trials– is an unbiased estimator of p.

• The standard deviation of the sampling distribution of is

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Large-Sample Confidence Interval for p

• Where:

• When n is large, we can use to approximate the value of p in the formula for .

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Example

• A sample of 500 consumers chosen at random shows that 265 are optimistic about the state of the economy:– Calculate a 90% confidence interval to

estimate the proportion of all consumers that are optimistic about the state of the economy.

Answer

567.0,493.0

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Determining Sample Size:Population Mean Estimation

• Where:– W is the desired width of the confidence interval.– The population standard deviation must be

estimated.– n is rounded up to ensure that the sample size will

be sufficient to achieve the desired confidence interval.

Example

• Footballs inflated to a mean pressure of 13.5 pounds:– Due to machine calibration, individual

footballs vary in pressure from 13.3 to 13.7 pounds.

– What sample size is necessary for a 99% confidence interval that is only 0.05 wide?

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Determining Sample Size:Binomial Probability

• Where:– W is the desired width of the confidence interval.– The population standard deviation is estimated by

p*q.– n is rounded up to ensure that the sample size will

be sufficient to achieve the desired confidence interval.

Example

• The probability that a consumer will feel optimistic about the economy is 0.53:– What sample size is necessary for a 90%

confidence interval that is only 0.05 wide?

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Confidence Interval Review

• Large-Sample Confidence Interval:

• Small-Sample Confidence Interval:

• Large-Sample Binomial Confidence Interval:

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Commonly Used Values of

z2

Confidence Level1001( )

2z2

90% .10 .051.64595% .05.0251.9699% .01.0052.575

Example

• Example:– We want to estimate the mean number of

unoccupied seats per flight for a major airline. Specifically, we want to construct a 90% confidence interval for the population mean.

– 225 flights sampled.– Sample mean is 11.6 empty seats.– Sample standard deviation is 4.1 seats.

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Solution

• Large sample size so use z-statistic rather than t-statistic.

• Formula:– where z-value is at 10/2=5% level. Z=1.645.– the population standard deviation is estimated by

the sample standard deviation.

– We are 90% confident that the mean number of unoccupied seats per flight (population mean) lies between 11.15 and 12.05 seats.

)05.12,15.11(45.06.11)225

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2

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Example

• Earnings per share example:– We want to estimate the earnings per share of a type

of stock so we ask five portfolio analysts what their projections are for the upcoming year. Specifically, we want to construct a 95% confidence interval for the mean projected earnings estimate for all analysts.

– 5 analysts sampled.– Sample mean is $2.63 per share.– Sample standard deviation is $0.72 per share.

Solution• Small sample size so use t-statistic rather than z-

statistic.• Formula:

– where t-value is at 5/2=2.5% level and there are 5-1=4 degrees of freedom. t=2.776.

– the population standard deviation is estimated by the sample standard deviation.

– We are 95% confident that the mean of all analysts’ earnings per share projections (population mean) for this type of stock is between $1.74 and $3.52.

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Example

• Smokers brand preference example:– Philip Morris wants to determine the proportion of

smokers who prefer Marlboro. Specifically, they want to construct a 95% confidence interval for the proportion of smokers in the smoking population that prefer Marlboro.

– 1000 smokers interviewed.– X, the number of smokers (out of 1000 sampled) that

prefer Marlboro is a binomial random variable.– 313 out of the 1000 smokers prefer Marlboro:

313.01000

313ˆ p 687.0313.01ˆ1ˆ pq

Solution

• Binomial distribution so use the following formula:

– where z-value is at 5/2=2.5% level. Z=1.96.– the population standard deviation is estimated by the

sample standard deviation in the formula above.

– We are 95% confident that the proportion of smokers (population proportion) that prefer Marlboro is between 28.4% and 34.2%.

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Determining Sample Size

• Population Mean Estimation:

• Binomial Probability:

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Population Mean Example

• We want to determine the sample size necessary such that a 95% confidence interval for the mean overdue amount for all delinquent accounts is within $5 of the population mean:– A previous sample of delinquent accounts shows a

standard deviation of $90.– Note that information concerning mean is NOT

necessary.

– Use 1245 as the sample size.

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Binomial Probability Example

• The probability that a consumer will choose brand X is 0.20. What sample size is necessary for a 95% confidence interval that is only 0.08 wide?

– Use 385 as the sample size.

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