Chapter 10, Problem 10.65RE

(a)

To determine

To Explain: that there sufficient evidence to conclude that eating a three-ounce dark chocolate bar every day will lower one’s systolic bold pressure and write the null and alternative hypotheses for this experiment.

Answer to Problem 10.65RE

  $\begin{array}{c}{H}_0:\mu =0\\ H_1:\mu <0\end{array}$

Explanation of Solution

Given:

  $\begin{array}{c}\overline{x}=-5\\ s=6\\ n=7\end{array}$

Claim: Mean difference is smaller than 0(lowered)

Assume:

  $\alpha =0.05$

The claim is either the null hypothesis or the alternative hypothesis .the null hypothesis statement is that the population mean is equal to the value given in the claim. If the null hypothesis is the claim, then the alternative hypothesis statement is of the null hypothesis.

  $\begin{array}{c}{H}_0:\mu =0\\ H_1:\mu <0\end{array}$

(b)

To determine

To Calculate: the test statistic and the p-value and there is sufficient evidence to reject null hypothesis.

Answer to Problem 10.65RE

Yes

Explanation of Solution

Formula used:

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

Calculation:

The sample size of $n=6$ is not enough large, therefore it cannot be use for the normal distribution. However, it can be use for the Student’s $t$ -distribution. The sampling distribution of the sample mean has mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}\cdot$

The Z-value is

  $\begin{array}{c}z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}\\ =\frac{-5-0}{6/\sqrt{7}}\\ =-2.20\end{array}$

The P-Value is the probability of getting a value more extreme or equal to the standardized test statistic $z$ , assuming that the null hypothesis is true, the probability using normal probability table

  $P=P\left(Z<-2.20\right)=0.0139$

If the P-value is lesser than the significance level $\alpha$ , then the alternative hypothesis is accepted and null hypothesis is rejected:

  $P<0.05\Rightarrow {\text{Reject H}}_0$

(c)

To determine

To Explain: the conclusion in plain language.

Answer to Problem 10.65RE

There is alternative hypothesis is accepted.

Explanation of Solution

There is no enough evidence to help the claim that eating a three-ounce dark chocolate bar every day will lower one’s systolic blood pressure.

(d)

To determine

To Explain: that any concerns about the design of this experiment and done anything differently.

Answer to Problem 10.65RE

Yes, increase the sample size.

Explanation of Solution

The sample size of $n=7$ is very small. So it would have more confidence in the results if the sample size was increased. Moreover, assume that the sampling distribution contained no outliners and no strong skewed but have no way to check if this was actually true.