2 Sample t Intervals and Tests

The Size of Quail Eggs

Problem: A researcher is trying to determine if the type of food located near nesting areas will affect the size of quail eggs. Quail egg sizes vary normally. The researcher records the size (in mm) of eggs found in nests near corn fields and those located near wheat fields. Summary statistics are shown below:

n sEggs found near corn fields: 24 22.82 .85Eggs found near wheat fields: 28 23.21 .74

Does this data support the researcher’s claim that the sizes of the eggs differ depending on the type of field located near the nests? Find a 95% confidence interval for the mean difference in size.

x

Step 1: 2 Sample t interval for means

Step 2: Assumptions:

• We are uncertain that the two samples are random samples.

• Populations vary normally.

• The samples appear to be independent.

Step 3:

(x1 −x2 ) ±t* s1

2

n1

+s2

2

n2

(22.82 −23.21) ±2.069.852

24+

.742

28

−.39 ± .46

(−.85,.07)

X1 = size of eggs near corn fieldsX2 = size of eggs near wheat fields degrees of freedom: use 23, one less than the smaller sample size

Step 4: We are 95% confident that the true mean difference in size between quail eggs found near corn fields and quail eggs found near wheat fields is between -.85 and .07mm. (We might have had an assumption violation, though.) 95% of confidence intervals constructed on random samples of these sizes using this technique will capture the true mean difference in quail egg sizes.

Problem: Using the same quail egg data, determine if quail eggs found in nests near wheat fields are larger than quail eggs found near corn fields.

n sEggs found near corn fields: 24 22.82 .85Eggs found near wheat fields: 28 23.21 .74

x

Step 1:

The mean difference in size between quail eggs found near corn fields and quail eggs found near wheat fields is zero.

The mean difference in size between quail eggs found near corn fields and quail eggs found near wheat fields is negative.

X1 = size of eggs near corn fieldsX2 = size of eggs near wheat fields

H0 : μ1 −μ2 =0

Ha : μ1 −μ2 < 0

Step 2: Assumptions:

We dealt with assumptions when we made the confidence interval so I will not repeat them here.

Step 3: t =(x1 −x2 )

s12

n1

+s2

2

n2

=22.82 −23.21

.852

24+

.742

28

=−1.750

with 23 degrees of freedom

Step 4:

Step 5: P-value = P(t <-1.750) = .04672.

Step 6: Reject H0, a test statistic this small will occur by chance alone only 4.7% of the time.

Step 7: We have evidence that the mean size of quail eggs found near wheat fields is greater than the mean size of quail eggs found near corn fields.

THE END