Question

3. In the USA the maximum acceptable level for exposure to microwave radiation is 10 μW/cm2 (microwatts per square centimetre). It is feared that a large TV transmitter may have raised the level of ambient microwave radiation above this level. Fourteen widely-spaced monitoring sites were selected among homes at roughly equal distances from the transmitter. For these sites, the mean radiation level was 10.95 μW/cm2, with a standard deviation of 1.24 μW/cm2.

(a) Do these results indicate that microwave radiation exposure is at an unacceptable level? Conduct an appropriate hypothesis test using α = 1%.

(b) Ifyouwereaparentwithyoungchildrenandyoulivednearthetransmitter,wouldyou want the significance level of the test to be 1% or 5%? Justify your answer. What significance level would you want if you were the owner of the transmitter?

(c) Provide a 90% confidence interval for the mean microwave radiation at this distance from the transmitter.

Answer 1

Since we are given a small sample size of 14, and it is not given clearly whether population standard deviation is known or not, we use t-distribution.

(a)H₀: = 10 ( level for exposure to microwave radiation is 10 μW/cm2 )

v/s H_a: > 10 (level for exposure to microwave radiation is greater than10 μW/cm2)

Our test statistic here is where xbar is the sample mean, s is the sample standard deviation and n is the sample size. Under the null hypothesis, our test statistic follows t distribution with n-1 degrees of freedom. i.e t_n-1,, since this is a one sided test

t= ( 10.95-10)/(1.24/ ) = 2.87

Critical Value = t_13,0.01 = 2.65. ( using percentage points for t distribution)

Since our test statistic > critical value , we have sufficient evidence at 1 % level of significance to reject the null hypothesis. Hence we can conclude that microwave radiation exposure is at an unacceptable level.

(b) If I were a parent with young children and I lived near the transmitter , I would want the significance level to be 1%. This is because = 0.01 means a greater accuracy. If we are not able to reject the null hypothesis at = 0.01 , we can conclude with even greater confidence that the microwave radiation exposure is at an acceptable level.

If I were the owner of the transmitter, I would want the significance level to be 5%. This is because   =0.01 implies greater accuracy. Maintaining this accuracy would be costlier than maintaining the accuracy when =0.05.

(c) (1-)% confidence interval for the mean microwave radiation is given by

( xbar - (), xbar + ( ) )

So, 90% confidence interval for the mean microwave radiation is given by

(10.95- t_0.05,13* 1.24/ , 10.95 + t_0.05,13* 1.24/ ) = (10.95- 1.771*0.33, 10.95+1.771*0.33)

= (10.36, 11.54) (t_0.05,13 = 1.771, using percentage points for t-

distribution)