Two 5-year girls, Alyse and Jocelyn, have been training to run a 1-mile race. Alyse's 1 mile time A is approximately Normally distributed with a mean of 13.5 and standard deviation of 2.5 minutes. Jocelyn's 1-mile time J is approximately Normally distributed with a mean of 12 minutes and a standard deviation of 1.5 minutes. Assuming A and J are independent random variables, what is the probability that Alyse has a smaller time than Jocelyn in a 1 mile race on a randomly selected day?

Refer the below solution for better understanding.

Step-by-step explanation:

Given :

Standard Deviation,

Calculation :

The null Hypothesis is

The alternative Hypothesis is

Now,

 --------------- (1)

putting the values of

in equation (1)

z = 3.5969

We reject the null hypothesis at

z-value is larger than the critical z.

3.5969 > 1.96

Therefore, there is enough statistical evidence to suggest that Alyse time is larger than Jocelyn in a 1 mile race on a randomly select day.

The probability that Alyse has a larger time than Jocelyn is 0.99983

Therefore, the probability that Alyse has a smaller time than Jocelyn is

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1.7 × 10⁻⁴

Step-by-step explanation:

The question relates to a two sample z-test for the comparison between the means of the two samples  

The null hypothesis is H₀:  μ₁ ≤ μ₂

The alternative hypothesis is Hₐ: μ₁ > μ₂

Where;

= 13.5

= 12

σ₁ = 2.5

σ₂ = 1.5

We set our α level at 0.05

Therefore, our critical z = ± 1.96

For n₁ = n₂ = 23, we have;

We reject the null hypothesis at α = 0.05, as our z-value, 3.5969 is larger than the critical z, 1.96 or mathematically, since 3.5969 > 1.96

Therefore, there is enough statistical evidence to suggest that Alyse time is larger than Jocelyn in a 1 mile race on a randomly select day and the probability that Alyse has a larger time than Jocelyn is 0.99983

Therefore;

The probability that Alyse has a smaller time than Jocelyn is 1 - 0.99983 = 0.00017 = 1.7 × 10⁻⁴.

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step-by-step explanation: