Question

Are very young infants more likely to imitate actions that are modeled by a person or simulated by an object? This question was the basis of a research study. One action examined was mouth opening. This action was modeled repeatedly by either a person or a doll, and the number of times that the infant imitated the behavior was recorded. Twenty-seven infants participated, with 12 exposed to a human model and 15 exposed to the doll. Summary values are shown below.

	Person Model	Doll Model
x	5.10	3.48
s	1.60	1.30

Is there sufficient evidence to conclude that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll? Test the relevant hypotheses using a 0.01 significance level. (Use a statistical computer package to calculate the P-value. Use μ_Person − μ_Doll. Round your test statistic to two decimal places, your df down to the nearest whole number, and your P-value to three decimal places.)

t=

df=

P-value=

State your conclusion.

We reject H₀. We do not have convincing evidence that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.

We do not reject H₀. We have convincing evidence that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.    

We reject H₀. We have convincing evidence that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.We do not reject H₀.

We do not have convincing evidence that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.

Answer 1

Solution:

Here, we have to use two sample t test for the difference between two population means assuming equal population variances. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H­₀: the mean number of imitations is not higher for infants who watch a human model than for infants who watch a doll.

Alternative hypothesis: H_a: the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.

H₀: µ₁ ≤ µ₂ versus H_a: µ₁ > µ₂

µ₁ = μ_Person

µ₂ = μ_Doll

Test statistic formula for pooled variance t test is given as below:

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

We are given

X1bar = 5.1

X2bar = 3.48

S1 = 1.6

S2 = 1.3

n1 = 12

n2 = 15

df = n1 + n2 – 2 = 12 + 15 – 2 = 25

α = 0.01

Critical value = 2.4851

(by using t-table)

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(12 – 1)*1.6^2 + (15 – 1)*1.3^2]/(12 + 15 – 2)

S_p² = 2.0728

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

t = (5.1 – 3.48) / sqrt[2.0728*((1/12)+(1/15))]

t = 1.62/0.5576

t = 2.9053

P-value = 0.0038

(by using t-table)

P-value < α = 0.01

So, we reject the null hypothesis

There is sufficient evidence to conclude that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.

We reject H₀. We have convincing evidence that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll.