Statistical Inference: Sampling, Confidence Intervals, Hypothesis Testing, Exams of Statistics

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Stat 224 Fall 2004 Nov. 2-3, 2004

Discussion 7

1 The Sampling Distribution of the Mean (σ unknown)

1.1 Theorem 6.

If X¯ is the mean of a random sample of size n taken from a normal population having the mean

μ and the variance σ^2 , and S^2 =

∑^ n

i=

(Xi − X¯)^2 n − 1

, then

t =

X¯ − μ S/

√

n

is a random variable having the t distribution with the parameter ν = n − 1.

2 Confidence Interval

(1 − α) × 100% Confidence Interval for μ (when σ is known or the same size is large):

( ¯x − zα/ 2 ·

σ √ n

, x¯ + zα/ 2 ·

σ √ n

)

(1 − α) × 100% Confidence Interval for μ (when σ is unknown):

( x ¯ − tn− 1 ,α/ 2 ·

s √ n

, x¯ + tn− 1 ,α/ 2 ·

s √ n

)

where s is the sample standard deviation.

2.1 Example (Exercise 7.80, page 276, Text)

While performing a certain task under simulated weightlessness, the pulse rate of 32 astronaut trainees increased on the average by 26.4 beats per minute with a standard deviation of 4.28 beats per minute. What can one assert with 95% confidence about the maximum error if ¯x = 26. 4 is used as a point estimate of the true average increase in the pulse rate of astronaut trainees performing the given task?

2.2 Example (Exercise 7.81, page 276, Text)

With reference to the preceding exercise, construct a 95% confidence interval for the true average increase in the pulse rate of astronaut trainees performing the given task.

1231 MSC [email protected] Ting-Li Lin

Stat 224 Fall 2004 Nov. 2-3, 2004

3 Error in Hypothesis Testing

H 0 is true H 0 is false Do not reject H 0 Correct decision Type II error Reject H 0 Type I error Correct decision

α = P (type I error) = P (H 0 is rejected when it is true) β = P (type II error) = P (H 0 is not rejected when it is false)

3.1 Example (Exercise 7.30, page 244, Text)

A process for making steel pipe is under control if the diameter of the pipe has a mean of 3. inches with a standard deviation of 0.0250 inch. To check whether the process is under control, a random sample of size n = 30 is taken each day and the null hypothesis μ = 3.0000 is rejected if X¯ is less than 2.9960 or greater than 3.0040. Find

(a) the probability of a Type I error;

(b) the probability of a Type II error when μ = 3.0050 inches.

3.2 Example (Exercise 7.51, page 257, Text)

With reference to the vacuum cleaner example on page 255, use Table 8 to find the probabilities of Type II errors for

(a) μ = 76.00;

(b) μ = 78.00.

4 One Sample t Test

Consider H 0 : μ = μ 0. And the test statistic:

t =

X¯ − μ S/

√

n

Ha Rejection Region for a Level α Test μ > μ 0 t ≥ tα,n− 1 μ < μ 0 t ≤ −tα,n− 1 μ 6 = μ 0 t ≥ tα/ 2 ,n− 1 or t ≤ −tα/ 2 ,n− 1

4.1 Example (Exercise 7.90, page 277, Text)

A laboratory technician is timed 20 times in the performance of a task, getting ¯x = 7.9 and s = 1. 2 minutes. If the probability of the Type I error is to be at most 0.05, does this constitute evidence against the null hypothesis that the average time is less than or equal 7.5 minutes?

1231 MSC [email protected] Ting-Li Lin