1. STAT 312 Chapter 7 - Statistical Intervals Based on a Single Sample
7.1 - Basic Properties of Confidence Intervals
7.2 - Large-Sample Confidence Intervals for a Population Mean and Proportion
7.3 - Intervals Based on a Normal Population Distribution
7.4 - Confidence Intervals for the Variance and Standard Deviation of a Normal Pop
Chapter 8 - Tests of Hypotheses Based on a Single Sample
8.1 - Hypotheses and Test Procedures
8.2 - Z-Tests for Hypotheses about a Population Mean
8.3 - The One-Sample T-Test
8.4 - Tests Concerning a Population Proportion
8.5 - Further Aspects of Hypothesis Testing

2. Consider a population random variable X, normally distributed.
“null value”
Null Hypothesis
Alternative
Hypothesis
Two-sided
Test statistic = ?
Random Sample

3. All have postively-skewed tails.
NOTE:
(n – 1)S2 = SS
All have postively-skewed tails.
Test statistic = ?

4. standard normal distribution
Notation: Recall… 1
standard normal distribution
Z ~ N(0, 1)
α/2
1 – α

5. standard normal distribution
Notation: Recall… 1
standard normal distribution
Z ~ N(0, 1)
Use Z-table or R
(see 4.3, slides 13 to 17 or Lec Notes, p ).
.025
.95

6. Notation:
α/2
1 – α
Suppose n = 12, i.e., df = 11.

7. Notation:
.025
.95
Suppose n = 12, i.e., df = 11.

9. .95 .025 = 3.82 = 21.92 Notation: Suppose n = 12, i.e., df = 11.
> qchisq(.025, 11, lower.tail = F)
[1]
> qchisq(.025, 11, lower.tail = F)
[1]
= 3.82
= 21.92

10. Test statistic:
Recall that, when hypothesis testing for one mean  of a normally-distributed random variable (with known ), the test statistic is.…
With “margin of error”
Confidence Interval for 
Acceptance Region for H0

11. Test statistic:
However, when hypothesis testing for one variance 2 of a normally-distributed random variable, the test statistic is given above!
Confidence Interval for 2
Acceptance Region for H0

12. Hubble Space Telescope
Launched April 1990
340 miles above Earth
15 orbits per day
Size of a bus
Cost = $2.5 billion
Expected to remain in operation till > 2020.
Primary mirror:
2.4 m in diameter
enables detection of objects 10–10 fainter than human eye can detect
thousands of high-res images
HOWEVER…

13. Hubble Space Telescope
First images were blurry…
Spiral Galaxy M100
but why???

14. Hubble Space Telescope

15. Hubble Space Telescope
Hubble’s main mirror being polished before installation. Its edges were polished very slightly too flat, leaving the telescope unable to focus perfectly.

16. Hubble Space Telescope
Corrective Optics Space Telescope Axial Replacement (COSTAR)
“spherical aberration”

17. Hubble Space Telescope

18. Hubble Space Telescope
Spiral Galaxy M100

19. Strong rejection of H0 at any .
Test statistic:
(Average human hair = 10–1 mm)
Design criteria:  = 10 nanometers (10–9 meters, or 10–6 mm)
Null Hypothesis
Alternative
Hypothesis
!!!!!!!
Random sample of n = 12 measurements around the perimeter yield s = 2200 nm.
Confidence Interval for 2
Strong rejection of H0 at any .
does not contain null value 2 = 100

20. (Average human hair = 10–1 mm)
Test statistic:
(Average human hair = 10–1 mm)
Design criteria:  = 10 nanometers (10–9 meters, or 10–6 mm)
Null Hypothesis
Alternative
Hypothesis
!!!!!!!
Random sample of n = 12 measurements around the perimeter yield s = 2200 nm.
Confidence Interval for 2
Confidence Interval for 
does not contain  = 10

21. (Average human hair = 10–1 mm)
Test statistic:
(Average human hair = 10–1 mm)
Design criteria:  = 10 nanometers (10–9 meters, or 10–6 mm)
Null Hypothesis
Alternative
Hypothesis
!!!!!!!
Random sample of n = 12 measurements around the perimeter yield s = 2200 nm.
p-value of sample
But recall…

22. Notation:
.025
.95
Suppose n = 12, i.e., df = 11.
= 3.82
= 21.92

23. Strong rejection of H0 at any .
Test statistic:
(Average human hair = 10–1 mm)
Design criteria:  = 10 nanometers (10–9 meters, or 10–6 mm)
Null Hypothesis
Alternative
Hypothesis
!!!!!!!
Random sample of n = 12 measurements around the perimeter yield s = 2200 nm.
p-value of sample
Strong rejection of H0 at any .