Question

A new species of bird, the Lilac-Breasted Raven, was recently discovered. A researcher is attempting to determine the average incubation time for this species' eggs to hatch. For compatison, the eggs of a common raven have eggs that take an average of μ = 19 days to hatch with a standard deviation of σ = 1.23 days. A sample of 13 eggs was collected and incubated. The length of time (in days) is listed in the table below:

17 19 22

21 17 16

18 18 16

19 17 20

18

At the 5% significance level, perform and interpret a hypothesis test to determine if the average incubation time for the Lilac-Breasted Raven is less than 19 days. (You may assume that the hatch times are normally distributed, and that σ = 1.23 days.)

Answer 1

As per the provided information:

The researcher wants to test whether the average incubation time for the Lilac-Breasted Raven is less than 19 days.

The total number of eggs, n is 13.

The population mean, is 19 days.

The population standard deviation, is 1.23 days.

The significance level, is 0.05.

The given data set of incubated eggs is given by,

17, 19, 22, 21, 17, 16, 18, 18, 16, 19, 17, 20, 18

The sample mean can be calculated as:

The null and the alternative hypothesis can be stated as:

The test is left tailed.

As the sigma value is known, so the appropriate test statistic to be used is z- test statistic.

Test statistic:

Hence, the value of the z-statistic is -2.02.

Decision of the test using critical value:

At the significance level 0.05, the left tailed critical value obtained from z-table is -1.645.

Since, the test statistic (-2.02) is less than the critical value (-1.645), so the decision is to reject the null hypothesis.

Decision of the test using p- value:

Using the standard normal table, the obtained p-value is given by,

Since, the p-value (0.0217) is less than the level of significance (0.05), so the decision is to reject the null hypothesis.

In conclusion, there is enough evidence to support that the average incubation time for the Lilac-Breasted Raven is less than 19 days at the significance level 0.05.