Download Presentation
## Testing a Hypothesis about means

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Testing a Hypothesis about means**• The contents in this chapter are from Chapter 12 to Chapter 14 of the textbook. • Testing a single mean • Testing two related means • Testing two independent means**Testing a single mean**• This chapter uses the gssft.sav data, which includes data for fulltime workers only. • The variables are: • Hrsl: number of hours worked last week • Agecat: age category • Rincome: respondents income**The left plot is a histogram of the number of hours worked**in the previous week for 437 college graduates The peak at 40 hours is higher than you would expect for a normal distribution. There is also a tail toward larger values of hours worked. It appears that people are more likely to work a long week than a short week. Example**The sample mean (47) is not equals to the sample median**(45). The distribution is right-skewed that is consistent with Sk=1.24 The distribution is not normal. How would you go about determining if 47 is an unlikely value if the population mean to be 40. Example basic statistics**Testing a single mean**The variance is unknown, • The statistic • The rejection region • The critical value of t can be found in many textbooks or SPSS.**Testing a single mean**• The standard error of the mean is • The t -statistic The 95% confidence interval of the difference is**The t-distribution**• The statistic used in the previous page follows a t-distribution with n-1 degrees of freedom. • This is a 2-tailed test. • The p-value is the probability that a sample t value is greater than 14.3 or less than -14.3. • The p-value in this example is less than 0.0005. • We can conclude that it’s quite unlikely that college graduates work a 40-hour on average.**The degree of freedoms in this test is 437-1=436. The t**distribution is very close to the normal. The critical values or confidence interval can be determined based on the normal population. Normal approximation**Hypothesis Testing**• The p-value is the probability of getting a test statistic equal to or more extreme than the sample result, given that the null hypothesis is true.**Testing a Hypothesis about Two related means**• We use the endoph.sav data set provided by the author. • Dale et al. (1987) investigated the possible role of in the collapse of runners. are morphine (吗啡)-like substances manufactured in the body. • They measured plasma (血浆)concentrations for 11 runners before and after they participated in a half-marathon run. • The question of interest was whether average levels changed during a run.**Testing a Hypothesis about Two related means**• This problem is recommended to use the paired-samples t test.**Testing a Hypothesis about Two related means**• The average difference is 18.74 that is large comparing with S.D.=8.3. • The 95% confidence interval for the average difference is (13.14, 24.33) that does not includes the value of o, you can reject the hypothesis. • An equivalent way or testing the hypothesis is the t test. The p-value is less than 0.0005, we should reject the hypothesis.**Testing a Hypothesis about Two related means**diff Stem-and-Leaf Plot Frequency Stem & Leaf 1.00 0 . 3 4.00 1 . 0127 5.00 2 . 00458 1.00 3 . 0 Stem width: 10.00 Each leaf: 1 case (s) Each difference uses only the first two digits with rounding.**Testing a Hypothesis about Two related means**• All the differences are positive. That is, the after values are always greater than the before values. • The stem-and-leaf plot doesn’t suggest any obvious departures from normality. • A normal probability plot, or Q-Q plot, can helps us to test the normality of the data.**Normal Probability Plot**• For each data point, the Q-Q plot shows the observed value and the value that is expected if the data are a sample from a normal distribution. • The points should cluster around a straight line if the data are from a normal distribution. • The normal Q-Q plot of the difference variable is nor or less linear, so the assumption of normality appears to be reasonable.**This section uses the gss.sav data set.**Consider the number of hours of television viewing per day reported by internet users and non-users. It is clear that both are not from a normal distribution. Testing Two Independent Means**Testing Two Independent Means**• We find that there are some problems in the data. • There are people who report watching television for 24 hours a day!! It is impossible. • Watch TV is not a very well-defined term. If you have the TV on while you are doing homework, are you studying or watching TV? • The observations in these two groups are independent. This fact implies “two independent means”.**Testing Two Independent Means**• Two sample means, 2.42 hours of TV viewing and 3.52 hours for those who don’t use the internet. A difference is about 1.1 hours. • The 5% trimmed means, which are calculated by removing the top and bottom 5% of the values, are 0.3 hours less for both groups than the arithmetic means. The trimmed means are more meaningful in this case study.**Testing Two Independent Means**• For testing the hypothesis • There are several cases:**Testing Two Independent Means**• In most cases the variances are unknown.**Testing Two Independent Means**• Output from t test for TV watching hours**Testing Two Independent Means**• In the output, there are two difference versions of the t test. • One makes the assumption that the variances in the two populations are equal; the other does not. • Both tests recommend to reject the hypothesis with a significant level less than 0.0005. • The two-tailed test used in the two tests. • Testing the equality of two variances will be given next section.**Testing Two Independent Means**• The 95% confidence interval for the true difference is • [0.77, 1.42] for equal variances not assumed, • [0.76, 1.42] for the equal variances assumed. • Both the intervals do not cover the value 0, we should reject the hypothesis.**F test for equality of Two Variances**• From the results below we have • The critical value is close to 1.00 that implies to reject the hypothesis that two populations have the same variance.**Levene’s test for equality of variances**• The SPSS report used the Levene’s test (1960) that is used to test if k samples have equal variances. • Equal variances across samples is called homogeneity of variance. • The Lenene’s test is less sensitive than some other tests. • The SPSS output recommends to reject the hypothesis.**Effect Outliers**• Some one reported watching TV for very long time, including 24 hours a day. • Removed observations where the person watch TV for more than 12 hours.**Effect Outliers**• The average difference between the two groups reduced from 1.09 to 1.05. • The conclusions do not have any change.**Introducing More Variables**• Let us consider more related variables to study on the TV watching time • Consider age, education, working hours.**Introducing More Variables**• We reject the hypothesis that in the population the two groups have the same average age, education, and hours. • Internet users are significantly younger, better educated, and work more hours per week.