Question

We are interested to know whether test animals respond to temperature by moving
toward a favoured temperature. Animals were placed in a temperature gradient marked
in units from zero (the starting point) in the middle of the gradient to positive at the
warm end and negative at the cold end. The mean position along the gradient was found
to be -1.352 units, the standard deviation was 12.267 units and n=500 individuals. Is
there a significant tendency to aggregate at either the cold or warm end of the gradient?
Start by setting up a null hypothesis in terms of the mean position along the gradient =
0. (This question is graded out of 12 marks)

Answer 1

The Central Limit Theorem

Let $X_1,X_2,...,X_n$ be a random sample from a distribution with mean $\mu$ and variance $\sigma^2.$ Then if $n$ is sufficiently large, $\bar{X}$ has approximately a normal distribution with $\mu_{\bar{X}}=\mu$ and $\sigma_{\bar{X}}^2=\sigma^2/n.$

If $n>30,$ the Central Limit Theorem can be used.

The provided sample mean is $\bar{X}=-1.352$ and the known population standard deviation is $\sigma=12.267,$ and the sample size is $n=500.$

The following null and alternative hypotheses need to be tested:

$H_0:\mu=0$

$H_1: \mu\not=0$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}$

The z-statistic is computed as follows:

$$z={\bar{X}-\mu_0 \over \sigma/\sqrt{n}}={-1.352-0 \over 12.267/\sqrt{500}}\approx-2.464469$$

Since it is observed that $|z|=2.464469>1.96=z_c,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$  is different than 0, at the 0.05 significance level.

Using the P-value approach: The p-value is $p=0.0137,$ and since $p=0.0137<0.05=\alpha,$ it is concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$  is different than 0, at the 0.05 significance level.

There a significant enough evidence to claim that animals respond to temperature by moving toward a favoured temperature, at the 0.05 significance level.