Question

Using the data from Lap 1 only, conduct a hypothesis test to determine if the mean time for completing a lap in races is the same as it is in practices.

Short answer

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Step-by-step solution

Step 1. Given two population means and known population standard deviations

The given data sets are from Terri Vogel's log book.

Step 2. To state the null and alternative hypothesis

Let${\mathrm{\mu }}_1$be the mean time to complete Lap 1 in races

Let ${\mathrm{\mu }}_2$be the mean time to complete Lap 1 in practices

The null and alternative hypothesis is given by,

localid="1653303177214" $$\begin{gathered} {\mathrm{H}}_0:{\mathrm{\mu }}_1={\mathrm{\mu }}_2{\mathrm{H}}_0:{\mathrm{\mu }}_1-{\mathrm{\mu }}_2=0 \\ {\mathrm{H}}_{\mathrm{a}}:{\mathrm{\mu }}_1\ne {\mathrm{\mu }}_2{\mathrm{H}}_{\mathrm{a}}:{\mathrm{\mu }}_1-{\mathrm{\mu }}_2\ne 0 \end{gathered}$$

Step 3. To find the name of the test

We have, ${\mathrm{H}}_{\mathrm{a}}:{\mathrm{\mu }}_1\ne {\mathrm{\mu }}_2$

Here, ${\mathrm{\mu }}_1$ is not equal to ${\mathrm{\mu }}_2$. Therefore, this is a Two – tailed test.

Step 4. To calculate the sample means

Let ${\overline{\mathrm{x}}}_1$be the sample mean to complete Lap 1 in races

Let ${\overline{\mathrm{x}}}_2$be the sample mean to complete Lap 1 in practices

Number of random samples in races,localid="1653304155165" ${\mathrm{n}}_1=20$

Number of random samples in races, ${\mathrm{n}}_2=15$

localid="1653306341815">x¯1=Sumofrandomsamplesn1=262720=131.35x¯2=Sumofrandomsamplesn2=207815=138.53

Step 5. To calculate sample standard deviation

Let ${\mathrm{s}}_1$be the sample standard deviations to complete Lap 1 in races

Let ${\mathrm{s}}_2$be the sample standard deviations to complete Lap 1 in practices