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AP4.41 The body’s natural electrical field helps wounds heal. If diabetes changes this field, it might explain why people with diabetes heal more slowly. A study of this idea compared randomly selected normal mice and randomly selected mice bred to spontaneously develop diabetes. The investigators attached sensors to the right hip and front feet of the mice and measured the difference in electrical potential (in millivolts) between these locations. Graphs of the data for each group reveal no outliers or strong skewness. The following computer output provides numerical summaries of the data.

Is there convincing evidence at the $\mathrm{Î\pm }=0.05$ level that the mean electrical
potential differs for normal mice and mice with diabetes?

Step by step solution

01

Given information

To determine the convincing evidence at the $\mathrm{Î\pm }=0.05$ level that the mean electrical potential differs for normal mice and mice with diabetes.

02

Explanation

Let, the given values as follows:
${\stackrel{¯}{x}}_1=13090$
${\stackrel{¯}{x}}_2=10022$
$s_1=4839$
$s_2=2915$
$n_1=24$
$n_2=18$
$\mathrm{Î\pm }=0.05$
The difference in mean is the assertion that has been made.
The null hypotheses as follows:
$H_0:{\mathrm{μ}}_1={\mathrm{μ}}_2$

And the alternative hypotheses as follows:

$H_1:{\mathrm{μ}}_1≠{\mathrm{μ}}_2$
Where,
The true mean electrical potential for normal mice is ${\mathrm{μ}}_1$.
Then the true mean electrical potential for mice with diabetes is ${\mathrm{μ}}_2$.

03

Explanation

Let's pretend that all of the test's requirements are met.
As a result, test statistics have the following value:
$t=\frac{{\stackrel{¯}{x}}_1-{\stackrel{¯}{x}}_2-\left({\mathrm{μ}}_1-{\mathrm{μ}}_2\right)}{\sqrt{\frac{32}{n_1}+\frac{L_2^2}{m_2}}}$

$=\frac{13090-10022-0}{\sqrt{\frac{{48399}^2}{24}+\frac{{2915}^2}{18}}}$

$=2.550$
Determine the degrees of freedom as follows:
$df=\min \left(n_1-1,n_2-1\right)$
$=\min (24-1,18-1)$
$=17$
Then, determine the $P$-value by the degrees of freedom as follows:
$0.02<P<0.04$
The null hypothesis is rejected when the P-value is less than or equal to the significance level. As a result,
$P<0.05⇒\text{Reject}{H}_0$
Therefore, there is convincing evidence that the mean electrical potential differs for normal mice and animals with diabetes at the $\mathrm{Î\pm }=0.05$ level.

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