Question

Improving the productivity of chickens. Farmers have discovered that the more domestic chickens peck at objects placed in their environment, the healthier and more productive the chickens seem to be. White string has been found to be a particularly attractive pecking stimulus. In one experiment, 72 chickens were exposed to a string stimulus. Instead of white string, blue-colored string was used. The number of pecks each chicken took at the blue string over a specified time interval was recorded. Summary statistics for the 72 chickens were x = 1.13 pecks, s = 2.21 pecks (Applied Animal Behavior Science, October 2000).

a. Estimate population mean number of pecks made by chickens pecking at blue string using a 99% confidence interval. Interpret the result.

b. Previous research has shown that = 7.5 pecks if chickens are exposed to white string. Based on the results, part a, is there evidence that chickens are more apt to peck at white string than blue string? Explain.

Short answer

a. A 99% confidence interval for the average number of pecks each 1chicken took at the blue string over a specified time is between 0.4534 and 1.8007.

b. It is concluded that there is evidence that chickens are more apt to peak at white string than blue string.

Step-by-step solution

Given information

Let n denotes the number of chickens, that is n=72

The sample mean is $\overline{x}=1.13$pecks

The sample standard deviation is$s=2.21$ pecks

Estimating the confidence interval for population mean 

a. To find a 99% confidence interval for true mean$\mu$

Here,

$\alpha =0.01$

$\frac{\alpha }{2}=0.005$

From the standard normal table, the critical value at$\alpha =0.01$is

$\begin{array}{rcl}{Z}_{\frac{\alpha }{2}}& =& Z_{0.005}\\ & =& 2.575\\ & & \end{array}$

Use the interval estimator,$\overline{x}\pm 2.57{\sigma }_{\overline{x}}$to calculate a confidence interval for the population parameter $\mu$.

Now,

$\overline{x}=1.13$,$s=2.21$and$n=72$into the interval estimator formula,

Then,

$\begin{array}{rcl}\overline{x}\pm 2.57{\sigma }_{\overline{x}}& =& \overline{x}\pm 2.575\left(\frac{s}{\sqrt{n}}\right)\\ & =& 1.13\pm 2.575\left(\frac{2.21}{\sqrt{72}}\right)\\ & =& 1.13\pm 0.6707\\ & =& \left(0.4593,1.8007\right)\\ & & \end{array}$

Therefore, 99% confidence interval for the average number of pecks each 1chicken took at the blue string over a specified time is between 0.4534 and 1.8007.

Interpretation for the given population mean 

b.

Since,

$\mu =7.5$

The 99% confidence interval is between 0.4594 and 1.8006

Here, it is observed that the population mean $\mu$not lies between the confidence intervals. Hence it is concluded that there is evidence that chickens are more apt to peak at white string than blue string.