Question

On your mark In track, sprinters typically use starting blocks because they think it will help them run a faster race. To test this belief, an experiment was designed where each sprinter on a track team ran a $50$-meter dash two times, once using starting blocks and once with a standing start. The order of the two different types of starts was determined at random for each sprinter. The times (in seconds) for 8 different sprinters are shown in the table.

a. Make a dotplot of the difference (Standing - Blocks) in $50$-meter run time for each sprinter. What does the graph suggest about whether starting blocks are helpful?

b. Calculate the mean difference and the standard deviation of the differences. Explain why the mean difference gives some evidence that starting blocks are helpful.

c. Do the data provide convincing evidence that sprinters like these run a faster race when using starting blocks, on average?

d. Construct and interpret a $90\%$confidence interval for the true mean difference. Explain how the confidence interval gives more information than the test in part (b).

Short answer

(a) The required dotplot is

(b) The mean and standard deviation respectively is $-0.1312,0.1193$

(c) The data is providing convincing evidence that sprinters run faster when using starting blocks.

(d) The confidence interval is$-0.2111,-0.0513$

Step-by-step solution

Part (a) Step 1: Given information

We need to find out the dotplot and the suggestion about whether the boxplot is helpful or not.

Part (a) Step 2: Explanation

We know that

The required dotplot is given as,

We can see that $7$of the $8$dots are to the left of $0$, indicating that the most of the differences are negative and, as a result, the majority of the timings with blocks are shorter than the durations with a standing start.

As a result, the graph reveals that starting blocks are beneficial (reduce time).

Part (b) Step 1: Given information

We need to find the mean difference and standard deviation.

Part (b) Step 2: Explanation

We know that

The mean is the division of the sum of values and no. of values.

So, the Mean is, ${\overline{x}}_d=-0.1312$

And the square root of the variance is used to compute the standard deviation.

So, the standard deviation is given as, $0.1193$

Because the sample mean of the differences is negative, it means that the mean time with blocks is less than the mean time with a standing start, implying that starting blocks are beneficial.

Part (c) Step 1: Given information

We need to find whether the data is providing convincing evidence or not.

Part (c) Step 2: Explanation

We know that

The null hypothesis asserts that the variables are unrelated, whereas the alternative hypothesis asserts that they are.

$$\begin{gathered} H_0:{\mu }_d=0 \\ {H}_{\alpha }:{\mu }_d<0 \end{gathered}$$

And expected frequencies are a product of row and column total divided by table total.

And The squared differences between the actual and predicted frequencies, divided by the expected frequency, make up the chi-square subtotals.

Therefore, the data is providing convincing evidence that sprinters run faster when using starting blocks.

Part (d) Step 1: Given information

We need to find the $90\%$confidence interval for the mean difference.

Part (d) Step 2: Explanation

We know that

The confidence interval is calculated as the sum or difference of the mean and margin of error.

So, confidence interval is $-0.2111,-0.0513$

The genuine mean difference (With Blocks-Standing start) is between $-0.2111$seconds and $-0.0513$seconds, according to $90\%$confidence.

Because the confidence interval gives a range of possible values for the mean difference, whereas the hypothesis test simply examines a claim about one single value for the mean difference, it provides more information than the hypothesis test.