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In hypothesis testing, a \(p - \)value is used to assist you support or reject the null hypothesis. The \(p - \)value is proof that a null hypothesis is false. The smaller the p-value, the more evidence there is that the null hypothesis should be rejected.

• The null hypothesis is rejected if the\(p - \)value is less than\(0.05\). This is substantial evidence against the null hypothesis.
• A large \(p( > 0.05)\) indicates that the alternative hypothesis is weak, hence the null hypothesis is not rejected.

(a)

Let the given be:

\(\begin{array}{c}n = 15\\t = 3.2\\\alpha = 0.05\\{H_0}:\mu = 20\\{H_a}:\mu > 20\end{array}\)

\(\begin{array}{c}df = n - 1\\ = 15 - 1\\ = 14\end{array}\)

The P-value is the chance of getting the test statistic's result, or a number that is more severe. The t-value in the row\(df = 14\)is the P-value, which is the number (or interval) in the column title of the student鈥檚 T table in the appendix:

\(0.001 < P < 0.005\)

Since the P-value is smaller than the significance level, so the null hypothesis is rejected.

\(P < 0.05 \Rightarrow {\rm{Reject }}{H_0}\)

The allegation that the genuine average reflectometer reading for a new type of paint under consideration is less than\(20\)is supported by appropriate evidence.

(b)

Let the given be:

\(\begin{array}{c}n = 9\\t = 1.8\\\alpha = 0.01\\{H_0}:\mu = 20\\{H_a}:\mu > 20\end{array}\)

\(\begin{array}{c}df = n - 1\\ = 9 - 1\\ = 8\end{array}\)

The P-value is the chance of getting the test statistic's result, or a number that is more severe. The t-value in the row\(df = 8\)is the P-value, which is the number (or interval) in the column title of the student鈥檚 T table in the appendix:

\(0.05 < P < 0.10\)

Since the P-value is smaller than the significance level, so the null hypothesis is rejected.

\(P > 0.01 \Rightarrow Not Reject {H_0}\)

The allegation that the genuine average reflectometer reading for a new type of paint under consideration is less than\(20\)is not supported by appropriate evidence.

(c)

Let the given be:

\(\begin{array}{c}n = 24\\t = - 0.2\\{H_a}:\mu > 20\end{array}\)

\(\begin{array}{c}df = n - 1\\ = 24 - 1\\ = 23\end{array}\)

Assume:\(\alpha = 0.01\)

The P-value is the chance of getting the test statistic's result, or a number that is more severe. The t-value in the row\(df = 23\)is the P-value, which is the number (or interval) in the column title of the student鈥檚 T table in the appendix:

\(P > 0.10\)

Since the P-value is smaller than the significance level, so the null hypothesis is rejected.

\(P > 0.01 \Rightarrow Not Reject {H_0}\)

The allegation that the genuine average reflectometer reading for a new type of paint under consideration is less than \(20\) is not supported by appropriate evidence.