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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 that,

\(\begin{array}{l}\sigma = 9\\{H_0}:\mu = 75\end{array}\)

\(\begin{array}{l}{H_a}:\mu < 75\\n = 25\\\overline x = 72.3\end{array}\)

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:

\(\begin{array}{l}z = \frac{{\overline x - {\mu _{\overline x }}}}{{{\sigma _{\overline x }}}}\\ = \frac{{72.3 - 75}}{{9/\sqrt {25} }}\\ \approx - 1.50\end{array}\)

This then means that the sample mean \(\overline x \) is \(1.5\) standard deviants below the null value of \(75\).

Given that,

\(\alpha = 0.002\)

The P-value is the probability of obtaining a value more extreme or equal to the standardized test statistic z, assuming that the null hypothesis is true.

Determine the probability using normal probability table in the appendix,

\(\begin{array}{l}P = P(Z < - 1.50)\\ = 0.0668\end{array}\)

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

\(P > 0.002\), so it is Fail to reject\({H_0}\).

There is not sufficient evidence to support the claim that the average drying time is less than \(75\).

Given that,

\(\begin{array}{l}\alpha = 0.002\\{\mu _a} = 70\end{array}\)

\(\beta \)is the type II error, which is the probability to fail to reject the null hypothesis, when the null hypothesis is false.

The critical value is the z-value in the normal probability table in the appendix corresponding to a probability of \(0.002\):

\(z = - 2.88\)

Then the rejection region contains all values smaller than \( - 2.88\).

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:

\(\begin{array}{l}\overline x = \mu + z\frac{\sigma }{{\sqrt n }}\\ = 75 - 2.88\frac{9}{{\sqrt {25} }}\\ \approx 69.816\end{array}\)

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

\(\begin{array}{l}z = \frac{{\overline x - {\mu _{\overline x }}}}{{{\sigma _{\overline x }}}}\\ = \frac{{69.816 - 70}}{{9/\sqrt {25} }}\\ \approx - 0.10\end{array}\)

Determine the probability of rejecting the null hypothesis using the normal probability table in the appendix:

\(\begin{array}{l}\beta (70) = P(Z > - 0.10)\\ = 1 - P(Z < - 0.10)\\ = 1 - 0.4602\end{array}\)

\(\beta (70) = 0.5398\)

Given that,

\(\begin{array}{l}\alpha = 0.002\\{\mu _a} = 70\\n = unknown\end{array}\)

\(\beta \)is the type II error, which is the probability to fail to reject the null hypothesis, when the null hypothesis is false.

The critical value is the z-value in the normal probability table in the appendix corresponding to a probability of \(0.002\):

\(z = - 2.88\)

Then the rejection region contains all values smaller than \( - 2.88\).

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:

\(\begin{array}{l}\overline x = \mu + z\frac{\sigma }{{\sqrt n }}\\ = 75 - 2.88\frac{9}{{\sqrt n }}\end{array}\)

If,

\(\begin{array}{l}{\mu _a} = 70\\\beta (70) = 0.01\end{array}\)

The critical value is the z-value in the normal probability table in the appendix corresponding to a probability of \(1 - 0.01 = 0.99\)(take the complement, because \(\beta \) is the probability of failing to reject the null hypothesis):

\(z = 2.33\)

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:

\(\begin{array}{l}\overline x = \mu + z\frac{\sigma }{{\sqrt n }}\\ = 70 + 2.33\frac{9}{{\sqrt n }}\end{array}\)

Add \(2.88\frac{9}{{\sqrt n }}\) to each of the equation:

\(75 = 70 + 5.21\frac{9}{{\sqrt n }}\)

Subtract \(70\) from each side,

\(5 = 5.21\frac{9}{{\sqrt n }}\)

Multiply each side by \(\sqrt n \):

\(5\sqrt n = 5.21(9)\)

Divide each side by \(5\),

\(\sqrt n = 9.378\)

Taking square on each side,

\(n = 87.946884\)

\(n \approx 88\)

Given that,

\(\begin{array}{l}\alpha = 0.01\\n = 100\end{array}\)

Type I error is the probability to reject the null hypothesis, when the null hypothesis is true.

The type I error when \(\mu = 76\) is then \(0\), because the null hypothesis is not true and thus the event of a type I error cannot occur.

\({\rm{P(type I error ) = 0}}\)

If \(\mu = 75\)then, the probability of the type I error is the significance level of the test ,

\(\begin{array}{l}{\rm{P(type I error ) = }}\alpha \\\alpha = 0.01\end{array}\)