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

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

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

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

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. 4.2-14). .025 .95

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

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

.95 .025 = 3.82 = 21.92 Notation: Suppose n = 12, i.e., df = 11. > qchisq(.025, 11, lower.tail = F) [1] 3.815748 > qchisq(.025, 11, lower.tail = F) [1] 21.92005 = 3.82 = 21.92

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

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

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 http://www.nasa.gov/mission_pages/hubble/spacecraft/index.html HOWEVER…

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

Hubble Space Telescope http://www.nasa.gov/mission_pages/hubble/spacecraft/index.html

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. http://www.spacetelescope.org/about/history/aberration_problem/

Hubble Space Telescope Corrective Optics Space Telescope Axial Replacement (COSTAR) “spherical aberration” http://hubblesite.org/the_telescope/nuts_.and._bolts/optics/costar/

Hubble Space Telescope http://www.nasa.gov/mission_pages/hubble/spacecraft/index.html

Hubble Space Telescope Spiral Galaxy M100

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

(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

(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…

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

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 .