9 HypothesisTesting SingleMeanAndSingleProportion (handout)

9 HypothesisTesting SingleMeanAndSingleProportion (handout) tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài...

Introduction to Hypothesis Testing Topic covers Topic Index Probability Point & Interval Estimates Hypothesis Testing – Single Mean & Single Proportion Hypothesis Testing – Difference of two means, two proportions Hypothesis Testing – One way ANOVA Hypothesis Testing – Non parametric Tests Bivariate Correlation, Linear Regression What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill of this city is µ = $42 population proportion Example: The proportion of adults in this city with cell phones is p = 68 The Null Hypothesis, H0 States the assumption (numerical) to be tested Example: The average number of TV sets in U.S Homes is at least three ( Is always about a population parameter, not about a sample statistic H0 : μ ≥ H0 : μ ≥ ) H0 : x ≥ The Null Hypothesis, H0 Begin with the assumption that the null hypothesis is true Similar to the notion of innocence until guilt is proven Always contains “=” , “≤” or “≥” sign May or may not be rejected The Alternative Hypothesis, HA Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S homes is less than ( H A: µ < ) Challenges the status quo Never contains the “=” , “≤” or “≥” sign May or may not be accepted Is generally the hypothesis that is believed (or needs to be supported) by the researcher Hypothesis Testing Process Claim: the population mean age is 50 (Null Hypothesis: Population H0: µ = 50 ) Now select a random sample Is x = 20 likely if µ = 50? If not likely, Suppose the sample REJECT Null Hypothesis mean age is 20: x = 20 Sample Reason for Rejecting H0 Sampling Distribution of x x 20 µ = 50 If H0 is true If it is unlikely that we would hypothesis that µ = 50 get a sample mean of this value then we reject the null if in fact this were the population mean… Level of Significance, α Defines unlikely values of sample statistic if null hypothesis is true Defines rejection region of the sampling distribution Is designated by α , (level of significance) Typical values are 01, 05, or 10 Is selected by the researcher at the beginning Provides the critical value(s) of the test Level of Significance and the Rejection Region Level of significance = H0: μ ≥ HA: μ < α Represents critical value α Rejection region is Lower tail test α H0: μ ≤ HA: μ > Upper tail test H0: μ = shaded α /2 HA: μ α /2 ≠3 Two tailed test p-value example Compare the p-value with Here: α If p-value < α , reject H0 If p-value ≥ α , not reject H0 p-value = 0228 α = 05 Since 0228 < 05, we reject the null hypothesis Example: Upper Tail z Test for Mean (σ Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month The company wishes to test this claim (Assume σ = 10 is known) Form hypothesis test: H0: μ ≤ 52 the average is not over $52 per month HA: μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Example: Find Rejection Region Suppose that α = 10 is chosen for this test Find the rejection region: Review: Finding Critical Value - One Tail Standard Normal Distribution Table (Portion) What is z given α = 0.10? 08 Z 07 1.1 3790 3810 1.2 3980 3997 1.3 4147 4162 09 3830 50 z Critical Value = 1.28 4015 4177 Example: Test Statistic Obtain sample evidence and compute the test statistic Suppose a sample is taken with the following results: n = 64, x = 53.1 (σ=10 was assumed known) Then the test statistic is: x−μ z= = ?? σ n Example: Decision Reach a decision and interpret the result: Do not reject H0 since z = 0.88 ≤ 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52 p -Value Solution Calculate the p-value and compare to α P( x ≥ 53.1 | μ = 52.0)    53.1 − 52.0  = P z <  10   64   Do not reject H0 Do not reject H0 since p-value = 1894 > α = 10 Example: Two-Tail Test (σ Unknown) The average cost of a hotel room in New York is said to be $168 per night A random sample of 25 hotels resulted in x = $172.50 and s = $15.40 Test at the α = 0.05 level (Assume the population distribution is normal) H0: μ = 168 HA: μ ≠ 168 Example Solution: Two-Tail Test H0: μ = 168 HA: μ ≠ 168  α = 0.05  n = 25  σ is unknown, so use a t statistic Do not reject H0: not sufficient evidence that true mean cost is different than  $168 Critical Value: t24 = ± 2.0639 Hypothesis Tests for Proportions Hypothesis Tests for Proportions Involves categorical values Two possible outcomes “Success” (possesses a certain characteristic) “Failure” (does not possesses that characteristic) Fraction or proportion of population in the “success” category is denoted by p Proportions Sample proportion in the success category is denoted by  p x number of successes in sample p= = n sample size When both np and n(1-p) are at least 5, p can be approximated by a normal distribution with mean and standard deviation  μP = p p(1 − p) σp = n ... Lesson plan 12 - Hµ Thu Hoµ - Chieng Sinh Upper Secondary School Eucalyptus Tree This snow gum sits atop Mount Spectacular in Victoria, Australia. Eucalyptus species are some of the most important trees in western Australian forests. In addition to their importance to the lumber industry, extracts from their bark are used in the manufacture of various dyes and drugs. Many species of eucalyptus are called gum trees for the resin that oozes from them. Cactus Date Palm Cultivated in arid, hot regions, the common date palm, Phoenix dactylifera, is valued for its fruit, the date, which is considered a symbol of wealth as well as an important food source. Frilled Lizard Lesson plan 12 - Hµ Thu Hoµ - Chieng Sinh Upper Secondary School Lesson plan 12 - Hµ Thu Hoµ - Chieng Sinh Upper Secondary School Handout: - Unit 9 - Listening Task 2 1. The talk examines .what they are and how they are formed. a. deserts b. animals c. sand 2. Desert is a hot, dry place. a. peaceful b. endless c. sandy 3. Desert is also a beautiful land of ……. and space. a. nature b. silence c. mountains 4 ……… and cause the growing of the world's deserts. a. Nature and animals b. Animals and humans c. Nature and humans 5. Rabbits contribute to the growing of deserts in Australia. They…… .every plant they can find. a. destroy b. eat c. keep After you listen: Tapescript Hello everyone. In today's talk I'm going to tell you something about deserts, what they are and how they are formed. A desert is a hot, dry, sandy place. A desert is also a beautiful land of silence and space. The sun shines, the wind blows, and time and space seem endless. Nothing is soft. The sand and the rocks are hard, and many of the plants, such as the cactus, have hard needles instead of leaves. The size and location of the world's deserts are always changing. Over millions of years, as climates change and mountains rise, new dry and wet areas develop. But within the last 100 years, deserts have been growing at a frightening speed. This is partly because of natural changes, but the greatest desert makers are humans. In the 19th century some people living in English colonies in Australia got rabbits from England. Today there are millions of rabbits in Australia, and they eat every plant they can find. The great desert that covers the centre of Australia is growing. Farming first began in the Tigris-Euphrates, but today the land there is a desert. In dry areas, people can plant crops on dry and poor land. When there are one or two very dry years, DESERT (1) … of the growth of deserts Measures to (2) ……the growth of deserts 3 4 5 6 Lesson plan 12 - Hµ Thu Hoµ - Chieng Sinh Upper Secondary School the plants die, and the land becomes desert. In developing countries, 90 percent of the people use wood for cooking and heat. They cut down trees for firewood. But trees are important. They cool the land under them and keep the sun off smaller plants. When leaves fall from a tree, they make the land richer. When the trees are gone, the smaller plants die, and the land becomes desert. Humans can make deserts, but humans can also prevent their growth. Algeria planted a green wall of trees across the edge of the Sahara to stop the desert sand from spreading. Mauritania planted a similar wall around its capital. Iran puts a thin covering of petroleum on sandy areas and plant trees. Other countries build long canals to bring water to the desert areas.