SOLUTION: The mean I.Q of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with mean I.Q 100 and standard deviation 12? (Test at 5% level of sig

Algebra ->  Statistics  -> Central-limit-theorem -> SOLUTION: The mean I.Q of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with mean I.Q 100 and standard deviation 12? (Test at 5% level of sig      Log On


   

Question 1201121: The mean I.Q of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with mean I.Q 100 and standard deviation 12? (Test at 5% level of significance)
Answer by math_tutor2020(3817) About Me  (Show Source):
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Population info
mu = population mean = 100
sigma = population standard deviation = 12

Hypothesis:
Null: mu = 100
Alternative: mu =/= 100
The "not equals" in the alternative hypothesis indicates we'll be doing a two-tailed test.

Sample info
xbar = sample mean = 99
n = sample size = 1600

Compute the test statistic
z = (xbar - mu)/(sigma/sqrt(n))
z = (99 - 100)/(12/sqrt(1600))
z = -3.33 approximately

Use a Z table such as this one
https://www.ztable.net/
to find that
P(Z < -3.33) = 0.00043
This is the approximate area under the standard normal curve to the left of z = -3.33
We're doing a two-tailed test, so we double this area value.
2*0.00043 = 0.00086
This is the approximate p-value.

The p-value 0.00086 is smaller than alpha = 0.05, so we reject the null.

We conclude that the sample of 1600 children having a sample mean IQ of 99 very likely could not have been drawn from a population with mean IQ 100 and standard deviation 12.