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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,

\(\begin{array}{l}n = 6\\t = - 2.3\\\alpha = 0.05\end{array}\)

Given claim: average is less than \(7.0\)

The claim i either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis state the opposite of the null hypothesis.

\({H_0}:\mu = 7.0\)

\({H_a}:\mu < 7.0\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the students T table in the appendix containing the t-value in the row

\(\begin{array}{l}df = n - 1\\df = 6 - 1\\df = 5\end{array}\)

You need to double the boundaries, because the test is two sided.

\(0.025 < P < 0.05\)

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

\(P < 0.05\), so it is reject \({H_0}\)

Given that,

\(\begin{array}{l}n = 15\\t = - 3.1\\\alpha = 0.01\end{array}\)

Given claim: average is less than \(7.0\)

The claim i either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis state the opposite of the null hypothesis.

\({H_0}:\mu = 7.0\)

\({H_a}:\mu < 7.0\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the students T table in the appendix containing the t-value in the row

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

You need to double the boundaries, because the test is two sided.

\(0.001 < P < 0.005\)

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

\(P < 0.01\), so it is reject \({H_0}\)

Given that,

\(\begin{array}{l}n = 12\\t = - 1.3\\\alpha = 0.05\end{array}\)

Given claim: average is less than \(7.0\)

The claim i either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis state the opposite of the null hypothesis.

\({H_0}:\mu = 7.0\)

\({H_a}:\mu < 7.0\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the students T table in the appendix containing the t-value in the row

\(\begin{array}{l}df = n - 1\\df = 12 - 1\\df = 11\end{array}\)

You need to double the boundaries, because the test is two sided.

\(P > 0.10\)

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

\(P > 0.05\), so it is fail to reject \({H_0}\).

Given that,

\(\begin{array}{l}n = 6\\t = 0.7\\\alpha = 0.05\end{array}\)

Given claim: average is less than \(7.0\)

The claim i either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis state the opposite of the null hypothesis.

\({H_0}:\mu = 7.0\)

\({H_a}:\mu < 7.0\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the students T table in the appendix containing the t-value in the row

\(\begin{array}{l}df = n - 1\\df = 6 - 1\\df = 5\end{array}\)

You need to double the boundaries, because the test is two sided.

\(P > 0.10\)

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

\(P > 0.05\), so it is fail to reject \({H_0}\)

Given that,

\(\begin{array}{l}n = 15\\\overline x = 6.68\\s/\sqrt n = 0.0820\end{array}\)

Let us assume \(\alpha = 0.05\)

Given claim: average is less than \(7.0\)

The claim i either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population mean is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis state the opposite of the null hypothesis.

\({H_0}:\mu = 7.0\)

\({H_a}:\mu < 7.0\)

Determine the vale of the test statistic:

\(\begin{array}{l}t = \frac{{\overline x - {\mu _0}}}{{s/\sqrt n }}\\t = \frac{{6.68 - 7.0}}{{0.0820}}\\t = - 3.90\end{array}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of the students T table in the appendix containing the t-value in the row

\(\begin{array}{l}df = n - 1\\df = 6 - 1\\df = 5\end{array}\)

You need to double the boundaries, because the test is two sided.

\(0.005 < P < 0.001\)

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

\(P < 0.05\), so it is reject \({H_0}\)