# 3-20 - Chapter 8-— Conﬁdence Interval Estimation ‘ b...

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Unformatted text preview: Chapter 8 -— Conﬁdence Interval Estimation - ‘ b) Estimate Us, the standard deviation of S. A <3"5 :. 0. 54,17 ’ j c) Estimate us, the mean of S. M5 : 8.5287 d) Put a give—or—take number on your estimate of us. SD 7 2 5‘ boa ,0 5 A 1 C 1 m ~ A5 Due 'ﬁﬁﬂmm Example 17: Sample Median n /0.000 Let Y denote the sample median of 15 numbers drawn at random with replacement from the following box, which contains a total of 25 balls: ' 099 69 Here ubox = 13 and obox = 7.2. a) Simulate 10,000 values of the median. Record the mean and the standard deviation of the simulated values. Describe the shape of the experimental histogram for the simulated values. A /\ MM: . 6;“: '2‘,» be H" shaped in \‘srfoﬂfqm * b) Estimate cm, the standard deviation of the median. A . GM: 3.0452 c) Estimate pm, the mean of the median. A . MM = 13.0458 d) Put a give-or—take number on your; estimate of w. 1 EM 1* “’7” - M V71 Io.ooo ‘ 94 ll (I aaaaaaaeaaaaaaaacataaeaaataatcccaaiai WWWWWE’J CHAPTER 9 - AN INTRODUCTION TO HYPOTHESIS TESTING 9.1 The Null Hypothesis and the Alternative Hypothesis In this chapter we will perform tests of significance, a branch of hypothesis testing, to decide whether the evidence provided by the data is strong enough to reject the null hypothesis. The research hypothesis, or the claim, is called the alternativehypothesis, HA. The chance explanation, or the explanation that nulliﬁes the claim, is the null hypothesis, Ho. ‘ In hypothesis testing, the null and alternative hypotheses are often expressed in terms of a population parameter, i.e. por ,u. Example 1: Proportion with Children Suppose it is claimed that in a certain state the proportion pof adults who have children is more than 0.6. a) Using standard statistical notation, write out. the alternative hypothesis. HA : ,0 > o. (a V ' , ‘ b) What is the many-valued null hypothesis that nulliﬁes HA? H O A‘ ’0 £- 0 - (p I c) What is the single—valued null hypothesis? HO: p:o.(o Example 2: Proportion of Smokers Suppose it is claimed that in a certain state the proportion pof adults who smoke is less than 0.875. a) Using standard statistical notation, write out the alternative hypothesis. HA: p < 0. 9'7 5 b) What is the many—valued null hypothesis that nulliﬁes HA? HO: p Z 0%75 c) What is the single-valued null hypothesis? Ho! [920.875 Chapter 9 — Introduction to Hypothesis Testing I 9.2 Tests for a Population Proportion Now let’s carry out the hypothesis test using a test statistic and a four-stage decision procedure. First we will use five—step simulation to test Ho and then we will use the central limit theorem to test H0. Example 3: Proportion with Children Suppose that in a sample survey of 640 adults, only 384 of them have children. Does this prove that the majority (i.e., more than 50%) of adults in America have children? Stage I. Formulate the Null Hypothesis and Alternative Hypothesis What is the alternative hypothesis? HA : p > CD. 5 What is the null hypothesis? Ho: P: 0.5 Stage II. Choose a Null-Hypothesis-Based Sampling Box Model Draw the appropriate box model for this scenario. Are the draws made with or without replacement? ‘ [1 [:13 ,1. [:7] ' i has Am‘Hren [C Clonal-less I7: QHD O/rQLQ5 m/ replaCen’ien‘t (we éimp l {pied +LH‘Y\3 5) Stage III. Estimate the P-Value of the Hypothesis Test The strength of evidence against H0 is measured by computing the P-value, or the observed signiﬁcance level of the test. If HA: p > p0, then P—value = P(simulated ,5 2 observed ,5 l p = p0). - If HA: p < p0, then ‘ P—value = P(simulated ,6 3 observed ,5 l p = p0). What is the event of interest for our example? 84 A P( ’02. 33V0,ch i 53:05) “0:041 Run 10,000 simulations to get the P—value. ’o—Vqlu: “A: O 96 _. i . é \ Chapter 9 - Introduction to Hypothesis Testing Stage IV. Decide Whether to Reject Ho The smaller the P—value, the stronger the evidence against H0. In this class we will compare the P-value to a ﬁxed level of significance, 9_.Q§_. l o If P s 0.05, the result is called statistically significant and we reject Ho. 0 If P > 0.05, the result is not statistically significant and we fail to reject Ho. 0 If P s 0.01, the result is called highly signiﬁcant, or statistically signiﬁcant at level 0.01. o If P g 0.001, the result is very highly significant, or statistically signiﬁcant at level 0.001. Is the P—value in our example signiﬁcant? P—q/Q‘UQ, _(_ 0.0ch ) so ou( C€§M\+ 2-: Vet?! lm‘eéhly sr‘ﬁxn ﬁFi’cq/‘l {- , What is your conclusion? We Cornell/told vélm‘é Hé I’MH hypo‘t‘kcfm‘s [3 «pal-58 “40/ fAPé/‘écfﬂaél‘i/ﬂl h\/Po'é‘}r\€§lk\$ ’(5 "£de > 0" Example 4: Proportion of Smokers Suppose that in a sample survey of 656 adults, it is found that 574 of them smoke. l l l l l l l l l D l 9 Q 0 Q Q 9 'X/ Does this prove thatﬁst 90%[of adults in America smoke? 57% g 0 ' a . «\$375 0 39; Q 0 Q g less—than 6, 6(0 " Stage I. Formulate the Null Hypothesis and Alternative Hypothesis What is the alternative hypothesis? HA 2 '0 < O . ‘1 What is the null hypothesis? ‘ l—lo'- P‘O-‘l Stage II. Choose a Null-Hypothesis—Based Sampling Box Model Draw the appropriate box model for this scenario. Are the draws made with or without replacement? , fol/23 11211 A L SMokCF XL non—snokc( ’1'; (06¢ alrawb w/(equCQMen/ILL .[qu‘Je el‘mpli‘ﬂl‘eolflm‘nﬁs) 97 ...
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3-20 - Chapter 8-— Conﬁdence Interval Estimation ‘ b...

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