Q. 33 - The Practice of Statistics for AP

Question

How is the hatching of water python eggs influenced by the temperature of the snake’s nest? Researchers randomly assigned newly laid eggs to one of three water temperatures: hot, neutral, or cold. Hot duplicates the extra warmth provided by the mother python, and cold duplicates the absence of the mother. Here are the data on the number of eggs and the number that hatched 

(a) Make a two-way table of temperature by outcome (hatched or not). Calculate the proportion of eggs in each group that hatched. The researchers believed that eggs would not hatch in cold water. Do the data support that belief? 

(b) Are the differences between the three groups statistically significant? Give appropriate evidence to support your answer 

Step-by-Step Solution

Answer

(a) The two-way table of temperature by the outcome is shown in the table

and yes the data support the belief.

(b) The differences between the three groups are not statistically significant.

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

The number of eggs not hatched is the number of eggs decreased by the number of eggs that hatched.

The proportions in each group is the number of eggs divided by the row total given in the last column:

We can see that the fraction is lowest in cold water, confirming the researcher's hypothesis. However, this percentage is still quite high.

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

Observed counts

The expected counts are the row total multiplied by the column total, divided by the sample size $187$.

The chi-square statistic is the sum of squared deviations (between observed and expected counts) divided by the expected count:

$$\begin{gathered} {\chi }^2=\frac{(16-18.6257{)}^2}{18.6257}+\frac{(11-8.3743{)}^2}{8.3743}+\frac{(38-38.631{)}^2}{38.631}+\frac{(18-17.369{)}^2}{17.369} \\ +\frac{(75-71.7433{)}^2}{71.7433}+\frac{(29-32.2567{)}^2}{32.2567} \\ =1.703 \end{gathered}$$

The interval for the P-value can then be in found in table C in the column title which have the $X^2$-value in the corresponding interval in the row with

$$\begin{gathered} d f=(c-1)(r-1) \\ =(3-1)(2-1) \\ =2 \end{gathered}$$

$P>0.25$

If the P-value is smaller than the significance level, then the null hypothesis is rejected

$P>0.05 \Rightarrow Fail to reject H_0$

There is not sufficient evidence that there is an association between the variables.