Question

Aminah wish to perform the hypothesis testing H0: μ =1 versus H1: μ <1 versus with α=0.10. . The sample size 25 was obtained independently from a population with standard deviation 10. State the distribution of the sample mean given that null hypothesis is true and find the critical value, then calculate the values of sample mean if she reject the null hypothesis. Finally, compute the p-value, if the sample mean is -2.

Answer 1

Solution

If many samples of size $n_i$ are taken from a population with mean $\mu$ and standard deviation $\sigma$, the distribution of sample means $\bar x_i$ will have a mean $\mu$ and a standard deviation of $\frac{\sigma}{\sqrt{n_i}}$

$$\ n_i = 25$$

$$\sigma = 10$$

$$\frac{\sigma}{\sqrt{\ n_i}} = \frac{10}{\sqrt{25}} = 2$$

$$\mu = 1$$

Therefore: $\bar x$ ~ $N( 1, 2)$

The test statistic

$$t = \frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n_i}}}$$

At the level of significance, $\alpha = 0.1$ (since this is a one tail test) and degrees of freedom $v=\ n_i - 1 = 24$ , the test statistic

$$t = 1.318$$

Since we are testing the alternative hypothesis that $\mu < 1$ ,

$$1.318 = \frac{1 - \bar x}{2}$$ $$\bar x = 1 - (1.318 * 2) = -1.636$$

Aminah rejects the null hypothesis if $\bar x < -1.636$

When the sample mean $\bar x = -2$

$$t = \frac{1--2}{2} = 1.5$$

when $t = 1.5$, and degrees of freedom $v=24$ the p-value is $= 0.0733$