Question

3) A researcher is interested in studying whether or not baseball players are more superstitious than people in general. The population has an average score of 4.3 (SD = 1.9) on the superstitious scale. A sample of 36 baseball players scored an average of 4.7 on the superstitious scale. What does the distribution of sample means for this scenario consist of? What would we conclude and explain your decision using the distribution of sample means in your explanation.

Answer 1

Solution

Back-up Theory

CENTRAL LIMIT THEOREM ……………………………………………………………. (1)

Let {X₁, X₂, …, X_n} be a sequence of n independent and identically distributed (i.i.d) random variables drawn from a distribution [i.e., {x₁, x_2, …, x_n} is a random sample of size n] of expected value given by µ and finite variance given by σ². Then, as n gets larger, the distribution of Z = {√n(Xbar − µ)/σ}, approximates the normal distribution with mean 0 and variance 1 (i.e., Standard Normal Distribution)

i.e., sample average from any distribution follows Normal Distribution with mean µ and variance

σ²/n ………………………………………………………………………………………….. (1)

Now to work out the solution,

Let X = superstitious score. We are given µ = 4.3 and σ = 1.9 ……………………………….. (2)

Part (a)

Vide (1) and (2), distribution of sample means for this scenario would be

N(4.3, 0.31²) Answer

[here n = 36 and so √n = 6 ]

Part (b)

To derive a conclusion from the given data. We will perform a z-test for µ = 4.3 as detailed below.

Claim: Baseball players are more superstitious than people in general.

Hypotheses:

Null H₀: µ = µ₀ = 4.3   Vs Alternative H_A: µ > µ₀

Test statistic:

Z = (√n)(Xbar - µ₀)/σ, where n = sample size; Xbar = sample average; σ = known population standard

      deviation.

= 6(4.7 – 4.3)/1.9

= 1.2632

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

Under H₀, Z ~ N(0, 1)

Critical value = upper α% point of N(0, 1).

p-value = P(Z > Zcal)

Using Excel Functions: Statistical NORMSINV and NORMSDIST,

Zcrit = 1.645 [assuming α = 0.05 i.e., 5% ]and

p-value = 0.1033

Decision:

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

Conclusion:

There is not sufficient evidence to support the claim that the Baseball players are more superstitious than people in general. Answer

DONE