**Testing means, part IIIThe two-sample t-test**

**One-sample t-test** Null hypothesis The population mean is equal to o Sample Null distribution t with n-1 df Test statistic compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

**Paired t-test** Null hypothesis The mean difference is equal to o Sample Null distribution t with n-1 df *n is the number of pairs Test statistic compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

**Comparing means** • Tests with one categorical and one numerical variable • Goal: to compare the mean of a numerical variable for different groups.

**Paired vs. 2 sample comparisons**

**2 Sample Design** • Each of the two samples is a random sample from its population

**2 Sample Design** • Each of the two samples is a random sample from its population • The data cannot be paired

**2 Sample Design - assumptions** • Each of the two samples is a random sample • In each population, the numerical variable being studied is normally distributed • The standard deviation of the numerical variable in the first population is equal to the standard deviation in the second population

**Estimation: Difference between two means** Normal distribution Standard deviation s1=s2=s Since both Y1 and Y2 are normally distributed, their difference will also follow a normal distribution

**Estimation: Difference between two means** Confidence interval:

**Standard error of difference in means** = pooled sample variance = size of sample 1 = size of sample 2

**Standard error of difference in means** Pooled variance:

**Standard error of difference in means** Pooled variance: df1 = degrees of freedom for sample 1 = n1 -1 df2 = degrees of freedom for sample 2 = n2-1 s12 = sample variance of sample 1 s22 = sample variance of sample 2

**Estimation: Difference between two means** Confidence interval:

**Estimation: Difference between two means** Confidence interval: df = df1 + df2 = n1+n2-2

**Costs of resistance to disease** 2 genotypes of lettuce: Susceptible and Resistant Do these differ in fitness in the absence of disease?

**Data, summarized** Both distributions are approximately normal.

**Calculating the standard error** df1 =15 -1=14; df2 = 16-1=15

**Calculating the standard error** df1 =15 -1=14; df2 = 16-1=15

**Calculating the standard error** df1 =15 -1=14; df2 = 16-1=15

**Finding t** df = df1 + df2= n1+n2-2 = 15+16-2 =29

**Finding t** df = df1 + df2= n1+n2-2 = 15+16-2 =29

**The 95% confidence interval of the difference in the means**

**Testing hypotheses about the difference in two means** 2-sample t-test

**2-sample t-test** Test statistic:

**Hypotheses**

**Null distribution** df = df1 + df2 = n1+n2-2

**Calculating t**

**Drawing conclusions...** Critical value: t0.05(2),29=2.05 t <2.05, so we cannot reject the null hypothesis. These data are not sufficient to say that there is a cost of resistance.

**Assumptions of two-sample t -tests** • Both samples are random samples. • Both populations have normal distributions • The variance of both populations is equal.

**Two-sample t-test** Null hypothesis The two populations have the same mean 12 Sample Null distribution t with n1+n2-2 df Test statistic compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

**Quick reference summary: Two-sample t-test** • What is it for? Tests whether two groups have the same mean • What does it assume? Both samples are random samples. The numerical variable is normally distributed within both populations. The variance of the distribution is the same in the two populations • Test statistic: t • Distribution under Ho: t-distribution with n1+n2-2 degrees of freedom. • Formulae:

**Comparing means when variances are not equal** Welch’s t test

**Burrowing owls and dung traps**

**Dung beetles**

**Experimental design** • 20 randomly chosen burrowing owl nests • Randomly divided into two groups of 10 nests • One group was given extra dung; the other not • Measured the number of dung beetles on the owls’ diets

**Number of beetles caught** • Dung added: • No dung added:

**Hypotheses** H0: Owls catch the same number of dung beetles with or without extra dung (m1 = m2) HA: Owls do not catch the same number of dung beetles with or without extra dung (m1m2)

**Welch’s t** Round down df to nearest integer

**Owls and dung beetles**

**Degrees of freedom** Which we round down to df= 10

**Reaching a conclusion** t0.05(2), 10= 2.23 t=4.01 > 2.23 So we can reject the null hypothesis with P<0.05. Extra dung near burrowing owl nests increases the number of dung beetles eaten.

**Quick reference summary: Welch’s approximate t-test** • What is it for? Testing the difference between means of two groups when the standard deviations are unequal • What does it assume? Both samples are random samples. The numerical variable is normally distributed within both populations • Test statistic: t • Distribution under Ho: t-distribution with adjusted degrees of freedom • Formulae:

**The wrong way to make a comparison of two groups** “Group 1 is significantly different from a constant, but Group 2 is not. Therefore Group 1 and Group 2 are different from each other.”

**A more extreme case...**