Question

In a study examining 131 collared flycatcher eggs, researchers measured various characteristics in order to study their relationship to egg size (assayed as egg volume, in mm3). These characteristics included nestling sex and survival. A single pair of collared flycatchers generally lays around 6 eggs per breeding season; laying order of the eggs was also recorded.

1. (a) Is there evidence at the α = 0.10 significance level to suggest that egg size differs between male and female chicks? If so, do heavier eggs tend to contain males or females? For male chicks, x = 1619.95, s = 127.54, and n = 80. For female chicks, x = 1584.20, s = 102.51, and n = 48. Sex was only recorded for eggs that hatched.

2. (b) Construct a 95% confidence interval for the difference in egg size between chicks that success- fully fledged (developed capacity to fly) and chicks that died in the nest. From the interval, is there evidence of a size difference in eggs between these two groups? For chicks that fledged, x = 1605.87, s = 126.32, and n = 89. For chicks that died in the nest, x = 1606.91, s = 103.46, n=42.

3. (c) Areeggsthatarelaidfirstasignificantlydifferentsizecomparedtoeggsthatarelaidsixth?For eggs laid first, x = 1581.98, s = 155.95, and n = 22. For eggs laid sixth, x = 1659.62, s = 124.59, and n = 20.

please can you add explanation, I would like to understand.

Thank you !

Answer 1

Note: Final answers are highlighted in colour.

a)

Given

For male chicks, = 1619.95, s₁ = 127.54, and n₁ = 80. For female chicks, = 1584.20, s₂ = 102.51, and n₂ = 48.

Hypothesis:

H₀:

H₁:

As both the sampes are large enough, n₁≥30 and n₂≥30

Standardized Test Statistic for Hypothesis Tests Concerning the Difference Between Two Population Means: Large, Independent Samples

Where D₀= 0 in the current case.

Z= = 1.74

p value for Z=1.74 is 0.081859

at α = 0.10, this difference is significant. so there is sufficient evidence at α = 0.10 significance level to suggest that egg size differs between male and female chicks.

If we take alternate hypothesis to be , the test will become a one tail test with the same Z value.

One tail value for Z=1.74 is 0.0409295, this is also significant at α = 0.10. This proves the alternate Hypothesis that .

Therefore we can conclude that heavier eggs tend to contain males.

b)

Given,

For chicks that fledged, = 1605.87, s₁ = 126.32, and n₁ = 89. For chicks that died in the nest, = 1606.91, s₂ = 103.46, n₂=42.

Confidence Interval for the Difference Between Two Population Means: Large, Independent Samples is

α = 0.05, α/2= 0.025.

Z_α/2= 1.95996

Therefore CI=

=(-41.878,39.798)

From this interval which is evenly balanced around 0, we can say there is no evidence of a size difference in eggs between these two groups.

c)

For eggs laid first, = 1581.98, s₁ = 155.95, and n1 = 22. For eggs laid sixth, = 1659.62, s₂ = 124.59, and n2 = 20.

Hypothesis:

H₀:

H₁:

Standardized Test Statistic for Hypothesis Tests Concerning the Difference Between Two Population Means: Large, Independent Samples

Where D₀= 0 in the current case.

Z=

p value for Z=-1.78987 is 0.0734748

if α = 0.10, this difference is significant. so there is sufficient evidence at α = 0.10 significance level to suggest that egg size differs between first laid egg and sixth laid egg.