Chapter 12.1, Problem 16E

To determine

To find: that these data is providing convincing evidence at the $\alpha \text{=0}\text{.05}$ .

Answer to Problem 16E

There is enough convincing proof that there is a linear relationship between time at the table and calories consumed in the population of toddlers.

Explanation of Solution

Given:

Output of the computer from a least squares regression analysis of the data

  

  

  

  

  

  $\begin{array}{l}n=\text{Sample size =20}\\ \alpha \text{=Significance level =0}\text{.01}\end{array}$

Claim: Slope is negative (reduction)

Formula used:

Test statistics

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}$

Calculation:

The calculation of the slope $b_1$ is mention in the row “Time” and in the column “Coef” of the mention of the computer output:

  $b_1=-3.0771$

The calculated standard deviation of the slope $S{E}_{b_1}$ is mention in the row “Time” and in the column “SE Coef” of the mention of the computer output:

  $S{E}_{b_1}=0.8498$

The null hypothesis or the alternative hypothesis is the statement the given claim. The null hypothesis is the statements that slope is zero. If the mention claim is the null hypothesis, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_0:{\beta }_1\ne 0\end{array}$

Test statistic:

  $\text{t=}\frac{b_1-{\beta }_1}{S{E}_{b_1}}=\frac{-3.0771-0}{0.8498}=-3.6210$

The P-value is the probability of getting the value of the test statistic. $df-n-2=20-2=18$

  $0.001=2\left(0.0005\right)<P<2\left(0.001\right)=0.002$

If the P-value is equal or lesser than to the significance level, then the null hypothesis should be rejected:

  $P<0.01\Rightarrow \text{Reject }{H}_0$

There is enough convincing proof that there is a linear relationship between time at the table and calories consumed in the population of toddlers.