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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,

\(\begin{array}{l}n = 52\\\overline x = 1.95\\s = 0.20\\\alpha = 0.01\end{array}\)

Given claim: Average is less than \(2\) sec.

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

\({H_0}:\mu = 2\)

The alternative hypothesis states the claim:

\({H_a}:\mu < 2\)

Since the sample is large, we can use the z-test.

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

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

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

\(\begin{array}{l}z = \frac{{1.95 - 2}}{{0.2/\sqrt {52} }}\\z \approx - 1.80\end{array}\)

The P-value is the probability of obtaining a value more extreme or equal to the standard deviation test statistic z, assuming that the null hypothesis is true. Determine the probability table in the appendix.

\(\begin{array}{l}P = P(Z < - 1.80)\\ = 0.0359\end{array}\)

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

\(P > 0.01\)\( \Rightarrow \)Fail to reject \({H_0}\)

There is not sufficient evidence to support the claim that the true average task time is less than two seconds.