Question

A company claims to have invented a hand-held sensor that can detect the presence of explosives inside a closed container. Law enforcement and security agencies are very interested in purchasing several of the devices if they are shown to perform effectively. An independent laboratory arranged a preliminary test. If the device can detect explosives at a rate greater than chance would predict, a more rigorous test will be performed. They placed four empty boxes in the corners of an otherwise empty room. For each trial they put a small quantity of an explosive in one of the boxes selected at random. The company’s technician then entered the room and used the sensor to try to determine which of the four boxes contained the explosive. The experiment consisted of 50 trials, and the technician was successful in finding the explosive 16 times. Does this indicate that the device is effective in sensing the presence of explosives, and should undergo more rigorous testing?

1. Test an appropriate hypothesis and state your conclusion.

2. Was your test one-tail upper tail, lower tail, or two-tail? Explain why you chose that kind of test in this situation.

3. Explain what your P-value means in this context.

Answer 1

1.

p is the proportion of the times that the device correctly detects the explosive.

There are four boxes .

So p=1/4= 0.25

The device is effective only if it can detect the explosives at a rate higher than the chance which is 0.25

Ho : p = 0.25 vs Ha : p > 0.25

Given that: x=16 , n=50

Here we need to use 1-proportion z test

Test statistic formula is,

To get p- value use Excel function,= NORMSDIST (z)

P-value = P(z>1.14)

= 1- P(z<1.14)

=NORMSDIST (1.14)

= 0.8729

P-value = 1-0.8729 = 0.1271

P-value = 0.1271

Decision Rule : If p- value is less than significance level (alpha) the we reject Ho.

When significance level () is not given we take that as 0.05

As p value (0.1271) is greater than significance level (0.05) , we fail to reject Ho.

Conclusion:

There is not enough evidence that the device can detect the presence of explosive inside the box.

2.

One tail , Upper tail

As Ha : p>0.25 ,

That is alternative hypothesis (claim) : the device is effective only if it can detect the explosive at a rate higher than chance.

3.

The p- value is 0.1271 that is approximately 13%

We could expect to find the explosives 16 or more times out of 50 is about 13% of the time .