Question

Section 1

Tennis players often spin a racquet to decide who serves first. Th e spun racquet can land with the manufacturer’s label facing up or down. A reasonable question to investigate is whether a spun tennis racquet is equally likely to land with the label facing up or down. (If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.) Suppose that you gather data by spinning your tennis racquet 100 times, each time recording whether it lands with the label facing up or down.

1.1.1

a. Describe the relevant long-run proportion of interest in words.

b. What statistical term is given to the long-run proportion you described in (a)?

c. What value does the chance model assert for the long-run proportion?

d. Suppose that the spun racquet lands with the label facing up 48 times out of 100. Explain, as if to a friend who has not studied statistics, why this result does not constitute strong evidence against believing that the spinning process is fair.

e. Is the result in (d) statistically significant evidence that spinning is not fair or is it plausible that the spinning process is fair?

Answer 1

Solution

Part (a)

The relevant long-run proportion of interest is the proportion of times the tennis racquet lands with the label facing up (or down). Answer 1

Part (b)

Statistical term given to the long-run proportion described in (a) is: Probability Answer 2

Part (c)

The value the chance model asserts for the long-run proportion is: ½ = 0.5 Answer 3

[The chance model is: ‘If the spun racquet is equally likely to land with the label facing in either direction, we say that the spinning process is fair.’]   

Part (d)

A non-technical way of explaining would be to say that the same racquet when spun another 100 times might as well land 50 times with label facing up. Yet another 100 spins may turn out to be 52 face-up. Thus, every time this process is repeated, the result is likely to be different. So, just once 48 which is less than 50 does not necessarily provide evidence against. Nor does 52; for that matter, neither 50 is going to be in favor.

However, if the result is ‘far’ away from 50, one need to go deeper. Then, the question is: what is ‘far’? That question can be answered only by an analytical method called ‘hypothesis testing’. Answer 4

Part (e)

To address the question, we will perform a full-fledged testing.

Let p be the true proportion of times racquet lands with the label facing up.

Claim :

The spinning process is fair.

Hypotheses:

Null H₀ : p = p₀ = ½ [claim] Vs Alternative H_A : p ≠ ½

Test Statistic:

Z = (p_hat - p₀)/√{p₀(1 - p₀)/n}

Where

p_hat = sample proportion and

n = sample size.

Calculations:

Given
p0	0.50
n	100
x	48
phat	0.48
Zcal	- 0.4000
α	0.05
Zcrit	1.9600
p-value	0.6892

Distribution, Significance Level, α Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper (α/2)% of N(0, 1), and

p-value = P(Z > | Zcal |)

Using Excel Function: Statistical NORMINV and NORMDIST these are found as shown in the above table.

Decision:

Since | Zcal | < Zcrit, or equivalently, since p-value > α, H₀ is accepted.

Conclusion :

There is enough evidence to suggest that the claim is valid.

Hence, we conclude that the spinning process is fair. Answer 5

DONE