Chapter 10.2, Problem 10.55E

(a)

To determine

To Explain: the state null and alternative hypotheses in terms of the mean difference in times for the population of all adults.

Answer to Problem 10.55E

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

Explanation of Solution

Given:

Claim: Mean difference is higher than 0 (faster)

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 find: the mean and standard deviation of the 21 observations and the subjects work faster with the scented mask and the mean improvement big enough to be important.

Answer to Problem 10.55E

Mean=0.9571

Standard deviation= 12.5481

Explanation of Solution

Formula used:

  $\overline{x}=\frac{{\displaystyle \sum x}}{n}$

Calculation:

The mean is

  $\overline{x}=\frac{{\displaystyle \sum x}}{n}$

  $\begin{array}{c}\overline{x}=\frac{-7.37-3.14+4.1+\mathrm{...}+11.90-6.26+6.67}{21}\\ =\frac{20.1}{21}\\ =0.9571\end{array}$

The sample contains $21$ data values:

  $n-21$

The variance is

  $\begin{array}{c}s=\sqrt{\frac{{\left(-7.37-0.9571\right)}^2+\mathrm{...}+{\left(6.67-0.9571\right)}^2}{21-1}}\\ =12.5481\end{array}$

(c)

To determine

To construct: a stem plot of the data and explain and about outliers.

Answer to Problem 10.55E

No outlier.

Explanation of Solution

Given:

  

Calculation:

Rounding all data values to the closet integer:

-7, -3, 4, -4, 19, -11, -1, 9, 3, -17, 14, -25, 16, -8, 9, -11, 25, -4, 12, -6, 7

Graph:

Stem plot

  

By seeing the stem plot, it is observed that there not appear to be outlines, because there are no having gaps in the stem plot. It is not showing any other problems that might higher inferences, because the stem plot is about symmetric and therefore not strongly skewed.

(d)

To determine

To Explain: that the state hypotheses improvement statistically significant.

Answer to Problem 10.55E

It is failed to reject the null hypothesis and the improvement is not statistically significant.

Explanation of Solution

Formula used:

  $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$

Calculation:

The value of the test statistic is

  $\begin{array}{c}t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}\\ =\frac{0.9571-0}{12.5481/\sqrt{21}}\\ =0.350\end{array}$

The P-value is the probability of getting the value of the test statistic, or a value more extreme; assume that the null hypothesis is true. $df=n-1=21-1=20:$

  $P>0.25$

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

  $P>0.05\Rightarrow {\text{Fail to reject H}}_0$

It is failed to reject the null hypothesis and the improvement is not statistically significant.