probability Question?

Question

A certain brand of electric bulbs has an average life of 300 hours with a standard deviation of 25. A random sample of 100 bulbs is tested. What is the probability that the sample mean will be less than 295?

Please try and be as specific as possible, I'm lost

Answer 1

Like Matthew said, the sample mean will be normally distributed with mean 300 and standard deviation 25/sqrt(100)=2.5. Now the probability that the sample mean is less than 295 is the probability that (the sample mean-300)/2.5 < (295-300)/2.5 (this is the same inequality, i just subtracted 300 from both sides and divided both sides by 2.5), which is the probability that
(the sample mean-300)/2.5 < -2.

There is a theorem saying that if you take a normal random variable, then subtract its mean from it and divide it by its standard deviation then you get a standard normal random variable (with mean 0 and standard deviation 1). The cumulative distribution function of the standard normal distribution is usually denoted by Phi(x), and it gives the probability that a standard normal random variable is less than x.

Thus, the above probability is equal to Phi(-2). Additionally, Phi(x) has the property that Phi(-x)=1-Phi(x), so the answer can be expressed as 1-Phi(2). This is useful because Phi(x) is usually given in printed tables for only positive values of x.

Answer 2

that would be done by the use of Z test formula :

z = ( raw score - mean score ) / standard deviation

here raw score = 295
mean score = 300
and std deviation = 25

substituting: z = (295 - 300)/ 25 = -5/25 = -0.2

using the standard normal curve you will find
-.2 with area 0.0793

because the z value you obtained is negative and you want to find the probability of less than 295

you will subtract the area from .5 which is the area of the
half curve

so, 0.5 - 0.0793 = 0.4207

and that is the answer

therefore its probability that the sample mean
will be less than 295

is 0.4207

Answer 3

the sample mean follows a normal distribution.

Sample mean ~ N( mean of the underlying normal , Variance / n)

Sample mean ~ ( 300 , 6.25 )

Answer 4

This question is a bit Statistics Lesson question.and formula.