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Conditions Required for a Valid Large-Sample
Confidence Interval for µ
1. A random sample is selected from the target population.
2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .
x
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Thinking Challenge• We have a random sample of customer order totals with an
average of $78.25 and a population standard deviation of $22.5.• A) Calculate a 90% confidence interval for the mean given a
sample size of 40 orders.• B) Calculate a 90% confidence interval for the mean given a
sample size of 75 orders.• C) Explain the difference in the 90% confidence intervals
calculated in A and B.• D)Calculate the minimum sample size needed to identify a 90%
confidence interval for the mean assuming a $5 margin of error.
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6.3
Confidence Interval for a Population Mean:
Student’s t-Statistic
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Small sample size problem for inference about
• The use of a small sample in making inference about presents two problems when we attempt to use the standard normal z as a test statistic.
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Problem 1
• The shape of the sampling distribution of the sample mean now depends on the shape of the population sampled.
• We can no longer assume that the sampling distribution of sample mean is approximately normal because the central limit theorem ensures normality only for samples that are sufficiently large.
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Solution to Problem 1
• We know that if our sample comes from a population with normal distribution the sampling distribution of sample mean will be normal regardless of the sample size.
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Problem 2
• The population standard deviation is almost always unknown. For small samples the sample standard deviaiton s provides poor approximation for .
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Solution to Problem 2(Small Sample with known)
Use the standard normal statistic
z x µ x
x µ n
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Solution to Problem 2(Small Sample with Unknown)
Instead of using the standard normal statistic
use the t–statistic
z x µ x
x µ n
t x µs n
in which the sample standard deviation, s, replaces the population standard deviation, .
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Student’s t-StatisticThe t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0.
The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.
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Degrees of Freedom
The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).
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zt
Student’s t Distribution
0
t (df = 5)
Standard Normal
t (df = 13)Bell-ShapedSymmetric‘Fatter’ Tails
The smaller the degrees of freedom for t-statistic, the more variable will be its sampling distribution.
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• We have a random sample of 15 cars of the same model. Assume that the gas milage for the population is normally distributed with a standard deviaition of 5.2 miles per galon.
• A) Identify the bounds for a 90% confidence interval for the mean given a sample mean of 22.8 miles per gallon.
• B) The car manufacturer of this particular model claims that the average gas milage is 26 miles per gallon. Discuss the validity of this claim using the 90% confidence interval calculated in A.
• C) Let a and b represent the lower and upper boundaries of 90% confidence intervl for the mean of the population. Is it correct to conclude that tere is a 90% probability that true population mean lies between a and b?
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Thinking Challenge
• In 1882 Michelson measured the speed of light. His values in km/sec and 299,000 substracted from them. He reported the results of 23 trials with a mean of 756.22 and a standard deviaition of 107.12.
• Find a 95% confidence interval for the true spped of light from these statistics.
• Interpret your result.