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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.

Given claim: the average braking distance reduced from the known average of \(120ft\).

The average is represented by the population mean \(\mu \).

The null hypothesis states that the population mean is equal to the value mentioned in the claim:

\({H_0}:\mu = 120ft\)

The alternative hypothesis states the

Given that, Normal distribution with

\(\begin{array}{l}\sigma = 10\\n = 36\end{array}\)

\(\overline x = 117.2\)

\(\alpha = 0.10\)

The sampling distribution of the sample mean \(\overline x \) has mean \(\mu \)and standard deviation \(\frac{\sigma }{{\sqrt n }}\).

The z-score is the value decreased by the mean, divided by the standard deviation:

\(\begin{array}{l}z = \frac{{\overline x - {\mu _{\overline x }}}}{{{\sigma _{\overline x }}}}\\ = \frac{{117.2 - 120}}{{10/\sqrt {36} }}\\ \approx - 1.68\end{array}\)

The P-value is the probability of obtaining a value more extreme or equal to the standardized test statistic z. determine the probability using normal probability table in the appendix.

\(\begin{array}{l}P = P(Z < - 1.68)\\ = 0.0465\end{array}\)

If the P-value is the smaller than the significance level \(\alpha \), then the null hypothesis is rejected.

\(P < 0.10\),so it鈥檚 reject the \({H_0}\).

The new design can be implemented, because there appears to be reduction in the true average braking distance for the new design.

Use the value,\({\mu _a} = 115\)

Determine the z-score corresponding to a probability of \(\beta = 0.10\) using the normal probability table in the appendix ,

\(z = - 1.28\)

The sampling distribution of the sample mean \(\overline x \) has mean \(\mu \)and standard deviation \(\frac{\sigma }{{\sqrt n }}\).

The corresponding sample mean is the population mean (of the null hypothesis) increased by the product of the z-score and the standard deviation:

\(\begin{array}{l}\overline x = \mu + z\frac{\sigma }{{\sqrt n }}\\ = 120 - 1.28\frac{{10}}{{\sqrt {36} }}\\ \approx 117.8667\end{array}\)

We will reject the null hypothesis if the sample mean is larger than \(117.8667\).

The z-score is the value decreased by the mean, divided by the standard deviation:

\(\begin{array}{l}z = \frac{{\overline x - {\mu _{\overline x }}}}{{{\sigma _{\overline x }}}}\\ = \frac{{117.8667 - 115}}{{10/\sqrt {36} }}\\ \approx 1.72\end{array}\)

Determine the probability that we fail to reject the null hypothesis using the normal probability table in the appendix,

\(\begin{array}{l}\beta = P(Z > 1.72)\\ = 1 - P(Z < 1.72)\\ = 1 - 0.9573\\ = 0.0427\end{array}\)

\(\beta = 4.27\% \)