Question

3.3) [8 pts] A sample of 29 randomly selected automobile owners were asked to keep a record of the kilometers they travel. The random sample showed an average of 28,350 kilometers and a standard deviation of 5,237 kilometers. Assuming the underlying population is normally distributed, test the hypothesis that automobiles are driven on average more than 28,000 kilometers per year at the 10% level. Hou = 28,000 Hu> 28,000 First, specify: "If {inequality} we fail to reject the null hypothesis." Then solve the inequality such that it has no variables in it, i.e., show your work and solve for the final values and finally decide if you reject or fail to reject.
please explain

Answer 1

\mu_0 = 28000

n = 29

$\\s = 5237 \\\bar{X} = 28350 \\\alpha = 0.1$

The test hypothesis is

$\\Null\;Hypothesis --> H_0: \mu = \mu_0 \\Alternate\;Hypothesis --> H_1: \mu > \mu_0$

This is a one-sided test because the alternative hypothesis is formulated to detect the difference from the hypothesized mean on the upper side

Now, the value of test static can be found out by following formula:

$\\t_0 = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \\t_0 = \frac{28350.0- 28000.0}{5237.0/\sqrt{29}} \\\\t_0 = 0.3599$

Using Excel's function =T.DIST.RT(t0,n-1), the P-value for t0 = 0.3599 in an upper-tailed t-test with 28 degrees of freedom can be computed as

Since P = 0.3608 > 0.1, we fail to reject the null hypothesis $H_0:\mu=28000.0 \;at\; \alpha = 0.1.$

Since the sample size is n = 29, degrees of freedom on the t-test statistic are n-1 = 29-1 = 28

This implies that

$t_{\alpha, n-1} = t_{0.1, 28} = 1.3125$

Since   we fail to reject the null hypothesis $H_0:\mu=28000.0 \;at\; \alpha = 0.1.$