Respuesta :
Answer:
Approximately 50% of the vehicle speeds were less than 42 mph
Step-by-step explanation:
The normal distribution is symmetric:
That is, 50% of the measures are below the mean and 50% of the measures are above the mean.
In this problem, we have that:
Mean = 42 mph
So
Approximately 50% of the vehicle speeds were less than 42 mph
Answer:
[tex] P(X<42)[/tex]
Since the sample size is large enough and we know that the distribution for X is approximately normal we can use the z score formula given by:
[tex] z = \frac{X-\mu}{\sigma}[/tex]
And for X =42 we got:
[tex] z = \frac{42-42}{15}= 0[/tex]
And we have this:
[tex] P(z<0) = 0.5[/tex]
So we expect abput 50% of the values below 42 mph.
Step-by-step explanation:
Previous concepts
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Solution to the problem
We assume that the random variable X represent the speed at impact and we know from the sample size of n =5000 that:
[tex] E(X)= 42, \sigma =15[/tex]
And we want to find thie probability:
[tex] P(X<42)[/tex]
Since the sample size is large enough and we know that the distribution for X is approximately normal we can use the z score formula given by:
[tex] z = \frac{X-\mu}{\sigma}[/tex]
And for X =42 we got:
[tex] z = \frac{42-42}{15}= 0[/tex]
And we have this:
[tex] P(z<0) = 0.5[/tex]
So we expect abput 50% of the values below 42 mph.
