Populations and samples

 Population: “a group of individual persons, objects, or items from which samples are taken for statistical measurement”  Sample: “a finite part of a statistical population whose properties are studied to gain information about the whole”

(Merriam-Webster Online Dictionary, http://www.m-w.com/, October 5, 2004)

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Examples

Population Samples  Students pursuing undergraduate engineering degrees

 Cars capable of speeds in excess of 160 mph.

 Potato chips produced at the Frito-Lay plant in Kathleen

 Freshwater lakes and rivers

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1 Basic statistics (review)

1. Sample Mean: n Xi X i 1 n  Example: At the end of a team project, team members were asked to give themselves and each other a grade on their contribution to the group. The results for two team members were as follows: Q S X Q = ______92 85 95 88 = ______85 75 X S 78 92

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Basic statistics (review)

1. Sample Variance: n n n 2 2 2 (Xi X) n Xi ( Xi ) S2 i 1 i 1 i 1 n 1 n(n 1)

 For our example:

Q S 2 SQ = ______92 85 95 88

85 75 2 SS = ______78 92

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2 Sampling distributions

 If we conduct the same experiment several times with the same sample size, the probability distribution of the resulting statistic is called a sampling distribution  Sampling distribution of the mean: if n observations are taken from a normal population with mean μ and variance σ2, then: ... x n 2 2 2 2 2 2 ... x n2 n

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Central Limit Theorem

 Given:  X : the mean of a random sample of size n taken from a population with mean μ and finite variance σ2,  Then,  the limiting form of the distribution of X Z ,n / n

is ______

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3 Central Limit Theorem

 If the population is known to be normal, the sampling distribution of X will follow a normal distribution.

 Even when the distribution of the population is not normal, the sampling distribution of X is normal when n is large.

 NOTE: when n is not large, we cannot assume the distribution of X is normal.

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Example:

The time to respond to a request for information from a customer help line is uniformly distributed between 0 and 2 minutes. In one month 48 requests are randomly sampled and the response time is recorded.

What is the probability that the average response time is between 0.9 and 1.1 minutes?

______2 ______

2 X ______X ______

z1 ______z2 ______

P(0.9 < X < 1.1) = ______

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4 Sampling distribution of the difference between two averages  Given:

 Two samples of size n1 and n2 are taken from two 2 populations with means μ1 and μ2 and variances σ1 and 2 σ2  Then, X1 X 2 1 2 2 2 2 1 2 X1 X 2 n1 n2 and

(X1 X 2 ) ( ) Z 1 2 2 2 1 2

n1 n2

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Sampling distribution of S2  Given:  S2 is the variance of of a random sample of size n taken from a population with mean μ and finite variance σ2,  Then,

2 n 2 2 (n 1)s (Xi X) 2 2 i 1 has a χ2 distribution with ν = n - 1

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5 χ2 distribution

χ2

2 2  χα represents the χ value above which we find an area of α, that is, for 2 2 which P(χ > χα ) = α.

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Example

 Look at example 8.10, pg. 256: μ = 3 σ = 1 n = 5

s2 = ______

χ2 = ______χ2

If the χ2 value fits within an interval that covers 95% of the χ2 values with 4 degrees of freedom, then the estimate for σ is reasonable.

(See Table A.5, pp. 755-756)

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6 Your turn …  If a sample of size 7 is taken from a normal population (i.e., n = 2 2 2 7), what value of χ corresponds to P(χ < χα ) = 0.95? (Hint: first determine α.)

χ2

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t- Distribution

 Recall, by CLT:

X Z / n

is n(z; 0,1)

 Assumption: ______

(Generally, if an engineer is concerned with a familiar process or system, this is reasonable, but …)

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7 What if we don’t know σ?

 New statistic: X T S/ n Where,

n n X 2 (X X) X i and S i i 1 n i 1 n 1 follows a t-distribution with ν = n – 1 degrees of freedom.

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Characteristics of the t-distribution

 Look at fig. 8.13, pg. 259  Note:  Shape: ______

 Effect of ν: ______

 See table A.4, pp. 753-754

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8 Using the t-distribution

 Testing assumptions about the value of μ Example: problem 8.52, pg. 265

 What value of t corresponds to P(t < tα) = 0.95?

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Comparing variances of 2 samples

 Given two samples of size n1 and n2, with sample means X1 and 2 2 X2, and variances, s1 and s2 …

Are the differences we see in the means due to the means or due to the variances (that is, are the differences due to real differences between the samples or variability within each samples)?

See figure 8.16, pg. 262

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9 F-distribution  Given: 2 2  S1 and S2 , the variances of independent random samples of size n1 2 2 and n2 taken from normal populations with variances σ1 and σ2 , respectively,  Then,

2 2 2 2 S1 / 1 2S1 F 2 2 2 2 S2 / 2 1 S2

has an F-distribution with ν1 = n1 - 1 and ν2 = n2 – 1 degrees of freedom.

(See table A.6, pp. 757-760)

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Example

 Problem 8.55, pg. 266

2 S1 = ______

2 S2 = ______

F = ______f0.05 (4, 5) = ______

 NOTE: 1 f1 ( 1, 2 ) f ( 2, 1)

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