91影视

The null hypothesis, denoted by H₀, is the claim that is initially assumed to be true (the 鈥減rior belief鈥� claim). The alternative hypothesis, denoted by H_a, is the assertion that is contradictory to H₀.

The null hypothesis will be rejected in favour of the alternative hypothesis only if sample evidence suggests that H₀ is false. If the sample does not strongly contradict H₀, we will continue to believe in the plausibility of the null hypothesis. The two possible conclusions from a hypothesis-testing analysis are then reject H0 or fail to reject H₀.

The hypothesis of interest are \({H_0}:\mu = 50\) versus \({H_a}:\mu < 50\)

Assumption that the sample is from normal population distribution with known standard deviation allows using test statistic:

\(z = \frac{{\overline X - {\mu _0}}}{{\sigma /\sqrt n }}\)

The \(n\) is big enough to assume this. This z statistic value is

\(\begin{array}{l}z = \frac{{52.7 - 50}}{{4.8/\sqrt {45} }}\\z = 3.77\end{array}\)

The critical value \({z_\alpha }\), for \(\alpha = 0.05\)(other level of significance could have been chosen), because of one sided testing is,

\(\begin{array}{l}{z_\alpha } = {z_{0.05}}\\{z_\alpha } = 1.645\end{array}\)

The fact ,

\(\begin{array}{l}z = 3.77 > 1.65\\z = {z_\alpha }\end{array}\)

Implies that the null hypothesis is rejected at \(0.05\) level of significance. The conclusion is that the true average exceed \(50\) miles.