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Unformatted text preview: Chapter 7.1 — Inference for population means, σ unknown
Inference for the population mean µ when the population standard deviation σ is unknown Chapter 7.1 — Inference for population means, σ unknown When the population standard deviation σ is unknown we have to estimate it ﬁrst based on the collected data using the sample standard deviation s n 1 s= (xi − x )2 ¯ n−1
i =1 Using s instead of σ adds more variability to the distribution of need a distribution with heavier tails. x −µ ¯
s √ n , so we The t distribution accounts for the additional variation by having heavier tails (see graph next page)
(tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 1 / 12 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 2 / 12 Chapter 7.1 — Inference for population means, σ unknown Chapter 7.1 — Inference for population means, σ unknown
Recall that the normal distribution is characterized by two parameters: µ (the mean) and σ (the standard deviation) The t distribution is characterized by a single parameter, the socalled 0.4 0.3 dnorm(data) dnorm(data) 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 −4 −2 0 data 2 4 −4 −2 0 data 2 4 “degrees of freedom” (short:df) As the degrees of freedom increase, the t distribution approaches the standard normal distribution N (0, 1) Why? As the sample size increases, s estimates σ more accurately because we have more information about the population standard deviation in our sample.
(tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 4 / 12 0.4 0.3 dnorm(data) dnorm(data) −4 −2 0 data 2 4 0.2 0.1 0.0 0.0 −4 0.1 0.2 0.3 0.4 −2 0 data 2 4 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 3 / 12 Chapter 7.1 — Inference for population means, σ unknown
Reading the t table (Table D) Chapter 7.1 — Inference for population means, σ unknown dt(data, 5) 0.0 0.1 0.2 0.3 −4 −2 0 data 2 4 t distribution is symmetric it is characterized by the degrees of freedom
(tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 5 / 12 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 6 / 12 Chapter 7.1 — Inference for population means, σ unknown
Finding critical values t ∗ for a t distribution (table d) Example: 1 95% percentile of a t distribution with df=5: t ∗ is the critical value such that the area to the right (uppertail probability) of t ∗ is equal to 0.05 (or 5%) Chapter 7.1 — Inference for population means, σ unknown
2 the 5% percentile for a t distribution with df=15 3 1 2 3 Find the quantiles the bound the middle 95% look down the “df column” (ﬁrst column on left) to 5 at the top of the table, ﬁnd the right tail (uppertail) probability of 0.05 the critical value t ∗ corresponds to where row and column intersect, which is t ∗ = 2.015 Note: t table works with the area above (right) while z table works with area below (left)
(tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 7 / 12 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 8 / 12 Chapter 7.1 — Inference for population means, σ unknown Chapter 7.1 — Inference for population means, σ unknown
Example: 226 survey: How long has the longest relationship you have been in lasted? A random sample of 40 students yielded a mean 27.9 months and a standard deviation of 25.6 months. Find a 95% conﬁdence interval for the mean length µ of all relationships. confidence intervals for µ when σ is unknown A (1 − α) · 100% conﬁdence interval for µ is given by s ∗ x ±t · √ ¯ n just change σ to s and z ∗ to t ∗ look up t ∗ corresponding to a t distribution with df = n − 1 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 9 / 12 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 10 / 12 Chapter 7.1 — Inference for population means when σ is unknown Chapter 7.1 — Inference for population means, σ unknown
What about assumptions for the t test? simple random sample (ensuring independence of observations) data coming from a normal distribution or suﬃciently large sample size for the CLT to apply Handout (t test examples) ——————————————————————————————– Note, the t distribution converges to the Standard Normal distribution as the sample size increases, that is, values of t ∗ will get closer to corresponding z ∗ values. For this reason, it is not uncommon to base conﬁdence intervals and hypothesis tests on the Standard Normal distribution for larger sample sizes. Typically a sample size of 30 or more is assumed in order to do so. (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 11 / 12 (tDistribution, inference unknown σ ) Introduction to Business Statistics I November 16, 2009 12 / 12 ...
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This note was uploaded on 12/31/2009 for the course FSD FSD taught by Professor Vinh during the Spring '09 term at ITT Tech Flint.
 Spring '09
 VINH

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