Stat 512 Day 9: Confidence Intervals (Ch 5) Open Stat 512 Java Applets page

Last Time – Tests of Significance 0. Define the parameter of interest 1. Check technical conditions 2. State competing null and alternative hypotheses about the population parameter of interest (in symbols and in words) H 0 : parameter = value H a : parametervalue parameter = population mean,  ; population proportion,  Statement of research conjecture

PP (a) Let  represent the proportion of the population who prefer to hear bad news first. Ho:  =.5 (equally likely to prefer bad news as good news) Ha:  >.5 (majority of population prefer to hear bad news first) (b) Let  represent the average time (in hours) third and fourth graders spend watching television. Ho:  = 2 (spent 2 hours per day on average) Ha:  > 2 (the population average is more than 2 hours per day) Don’t forget the parameter! Always the equality Reflects the actual research conjecture

PP (c) Let  1 represent the proportion of all parolees in the population who receive a literature course who commit a crime within 30 months of parole. Let  2 represent the proportion of all parolees in the population without a literature course who commit a crime within 30 months of parole. Ho:  1 =  2 (no difference in the rate at which these populations commit a subsequent crime) Ha:  1 ≠  2 (there is a difference in the subsequent crime rate in these two populations)

Last Time – Tests of Significance 0. Define the parameter of interest 1. Check technical conditions 2. State competing null and alternative hypotheses about the population parameter of interest (in symbols and in words) Assume H 0 true, sketch picture of sampling distribution 3. Calculate test statistic 4. Determine p-value 5. State conclusion (reject or fail to reject the null hypothesis), translate back into English

Interpreting p-value P-value = probability of observing sample data at least this extreme when the null hypothesis is true due to “chance” (random sampling) alone  How often would we get data “like this” if the null was true  “Like this” is determined by the alternative If p-value is small  evidence against H 0 If p-value is large  lack of evidence against H 0  We don’t get to support the null! Guilty! Not Guilty! Can we prove the dice are fair?

Example? I love all chocolate ice cream  Have yet to find a choc ice cream don’t like… Data are behaving as expected based on that initial belief… have no reason to doubt it…  What if I find one that I don’t like? Specify alternative as what hoping to show…

Determining the p-value (PP) If Ha: parameter  > value, the p-value is probability above, P(Z>z) If Ha: parameter ≠ value, the p-value is 2P(Z>|z|), “two-sided” If Ha: parameter < value, the p-value is probability below P(Z<z)

State hypotheses before see data Social science research has established that the expression “absence makes the heart grow fonder” is generally true. Do you find this result surprising? Surprising 4Not Surprising 17 “out of sight, out of mind”… Surprising 0Not Surprising 22

Special cases: When want to know if parameter differs from hypothesized value, use two-sided H a  Doubles the p-value From now on, when working with quantitative data, will use the t distribution to find p-value  df = n – 1 (technology) Level of significance: May decide from the very start how low p-value will have to be to convince you, e.g.,.01,.05  Then if p-value < , say result is statistically significant at that level, e.g.,.05 or 5%

Example 4 Let  represent the ratio used by American Indians H 0 :  =.618 (used same ratio on average) H a :  ≠.618 (ratio used by American Indians differs) t = 2.05 with df = 19 P-value =.054 Weak evidence against H 0 Not overwhelmingly convincing that the mean ratio used by American Indians differs from.618. BUT this procedure probably not valid with these data since the sample size was small (n = 20 < 30) and it does not appear that the population distribution follows a normal distribution. Would need another analysis tool… If  =.618, how often would we find a sample mean at least as extreme.661 in a random sample?  =.618 =.661

The next question: People turn to the right more than half the time… the average healthy temperature of an American adult is not 98.6 o F… less than 51% of Brown athletes are women… Tests of significance have only told us what the parameter value is not, well what is it?

Example 1: Kissing the Right Way If more than half the population turns to the right, how often is it? 2/3 of the time? ¾ of the time? 70% of the time?

Plausible values for  H 0 :  =.50? two-sided p-value =.0012 H 0 :  = 2/3 two-sided p-value =.5538 H 0 :  =.70 two-sided p-value =.1814 H 0 :  =.75 Two-sided p-value =.0069  Sample proportions  ( ) Observed sample proportion

What are the plausible values of  ? The plausible values of  are those for which the two-sided p-value >.05 z = -2 z = 2 Empirical rule: 95% of sample proportions are within 2 standard deviations of  In 95% of samples,  should be within 2 standard deviations of

General Strategy To estimate population proportion, calculate sample proportion and look 2 standard deviations in each direction Standard “error” An approximate “95% confidence interval for  ”

In general A C% confidence interval (100-C)/2 90%.05 z* = %.005 z*=2.58 C%

Example 2: NCAA Gambling  = proportion of all male NCAA athletes who participate in some type of gambling behavior = sqrt(.634(1-.634)/12651) We are 95% confident that , the probability that an NCAA Div I athlete gambles, is between.626 and.642. Between 62.6% and 64.2% of Div I athlete gamble.

Example 3: Body Temperatures Let  = average body temperature of a healthy adult We are 95% confident that the mean body temperature of healthy adults is between o F and o F (assuming this sample was representative) What if we only asked women?

Example 4: NCAA Gambling cont What sample size is needed if we want the margin of error to be.01, with 95% confidence  With qualitative data, if don’t have a prior guess for , then use.5 – this guarantees margin of error is not larger…  If n is non-integer, always round up to the nearest integer

Example 5: What do we mean by “confidence”? Determine confidence interval using your sample proportion of orange candies… Did everyone obtain the same interval? Will everyone’s interval capture  ?

Interpretation of confidence What is the reliability of this procedure…  Assuming the Central Limit Theorem applies, how often will a C% confidence interval succeed in capturing the population parameter  To explore this, have to pretend we know the population parameter

Interpretation of confidence What if I put all of these confidence intervals into a bag and randomly selected one? The confidence level is a probability statement about the method, not individual intervals  If we had millions of intervals…  Sample size?

For Next Time Finish “surveys” on BB PP using applet HW 5  Problems 2 and 3… Review sheet and suggested problems have been posted on line