If the true level is 4.1, would it be unusual for a 24-hour measurement to indicate that the level is safe?

Question

Radon is a naturally occurring radioactive substance that is found in the ground underneath many homes. Radon detectors are often placed in homes to determine whether radon levels are high enough to be dangerous. A radon level less than 4.0 picocuries is considered safe. Because levels fluctuate randomly, the levels measured by detectors are not exactly correct, but are instead normally distributed. It is known from physical theory that when the true level is 4.1 picocuries, the measurement made by a detector over a one-hour period will be normally distributed with mean 4.1 picocuries and standard deviation 0.2 picocurie.

If a measurement is made for 24 hours, the mean will still be 4.1 picocuries, but the standard deviation will be only 0.04 picocurie. What is the probability that a 24-hour measurement will be below 4.0?

Answer 1

Hi Brianna,

The question is asking you to run a significance test to figure out if the results of the study differ significantly enough from the population value to determine if the test demonstrates a high probability that the population value is either wrong or has changed.

The null hypothesis (H₀) says that the test did not show a significant enough difference from the predicted value, and so the original value is likely still true.

The alternative hypothesis (H₁) says that the test value differs significantly enough from the predicted value that the predicted value is likely false.

We know that the mean true value, which can be interpreted as the mean population value, is equal to 4.1.

μ = 4.1

The problem gives us the standard deviation for both an hour and a 24 hour period, but we only care about the 24 hour period, in this case, which is 0.04.

σ = 0.04

And we are given that the test value is 4.0 and is it asking us the probability of having a test value less than 4.

x̅ = 4.0

P(x̅ < 4)

The "less than" tells us that this is a one-tailed test.

To figure the number of standard deviations 4 is from 4.1 we use the following formula

(μ - x̅) / σ = Z

Substitute for known values,

(4.1 - 4.0) / 0.04 = 2.5

Z = 2.5

We are looking for values less than 2.5 standard deviations from the population mean or,

P(Z < -2.5)

Now we go to our z table to figure out the probability of a one-tailed test being greater than 2.5 standard deviations from the population mean.

From the table, we see that

P(Z < -2.5) = 0.0062 = 0.62%

Therfore,

P(x̅ < 4) = 0.62%

I hope that helped!