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 test. 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? Test an appropriate hypothesis and state your conclusion. Find and interpret a 90% confidence interval.

Example of a proportional confidence interval formula:

CI99 = p + or - (2.58)[√(pq/n)]
...where p = x/n, q = 1 - p, and n = sample size.
Note: + or - 2.58 represents 99% confidence interval.

You will need to find the z-value for the 90% interval in the problem.

For p in your problem: 16/50 = .32
For q: 1 - p = 1 - .32 = 68
n = 50

I let you take it from here.

Correction: q = .68

To determine if the hand-held sensor is effective in sensing the presence of explosives, we can conduct a hypothesis test and calculate a confidence interval.

Hypothesis Test:
Null Hypothesis (H0): The hand-held sensor is not effective in sensing the presence of explosives.
Alternative Hypothesis (Ha): The hand-held sensor is effective in sensing the presence of explosives.

To test the hypothesis, we will use the binomial distribution since the technician either succeeded or failed in finding the explosive in each trial.

Using the binomial distribution, we can calculate the probability of getting 16 or more successful trials out of 50 trials if the device is not effective. If this probability is low, we can reject the null hypothesis and conclude that the device is effective.

Calculating the probability:
Assuming the null hypothesis is true, the probability of getting 16 or more successful trials out of 50 can be calculated as follows:

P(X >= 16) = 1 - P(X <= 15)

where X follows a binomial distribution with n = 50 and p = probability of success (unknown).

To calculate this probability, we need to know the true probability of success for this device. Since this information is not provided in the question, we cannot perform the hypothesis test.

Confidence Interval:
However, we can calculate a confidence interval to estimate the true probability of success of the device.

Using the Wilson score interval formula for a binomial proportion, we can calculate a 90% confidence interval for the true probability of success. The formula is:

CI = (p̂ + z^2/(2n) ± z * sqrt((p̂*(1-p̂) + z^2/(4n))/n)) / (1 + (z^2/n))

where p̂ is the sample proportion (16/50) and z is the z-score corresponding to a 90% confidence level (approximately 1.645).

Calculating the confidence interval:
CI = (0.32 + (1.645^2/(2*50)) ± 1.645 * sqrt((0.32*(1-0.32) + 1.645^2/(4*50))/50)) / (1 + (1.645^2/50))

Simplifying the equation provides the 90% confidence interval:

CI = (0.1356, 0.5044)

Interpretation:
The 90% confidence interval for the true probability of success of the hand-held sensor is approximately 13.56% to 50.44%. This means that we can be 90% confident that the true probability of success lies within this range.

However, since we couldn't perform the hypothesis test without knowing the true probability of success, we cannot make a conclusion about the effectiveness of the device based solely on the provided information. Further investigation and experiments are required.

To determine if the hand-held sensor is effective in sensing the presence of explosives, we can conduct a hypothesis test. The null hypothesis (H₀) would be that the device is not effective, and the alternative hypothesis (H₁) would be that the device is effective.

Let's assume that the technician randomly guesses which box contains the explosive in each trial. Under the null hypothesis, the probability of success (finding the explosive) would simply be 1/4 or 0.25.

Using statistical inference, we can use the binomial distribution to calculate the probability of getting the observed number of successes (16 out of 50 trials) if the null hypothesis is true. If this probability is very low, it would suggest that the device is indeed effective in sensing the presence of explosives.

Using a significance level of 0.05, we can perform a one-sided binomial test to test our hypothesis. In this case, we want to see if the observed number of successes (16) is significantly greater than what we would expect by chance (0.25).

We can calculate the p-value, which is the probability of observing 16 or more successes out of 50 trials if the null hypothesis is true. If the p-value is less than 0.05, we can reject the null hypothesis in favor of the alternative hypothesis.

To find the 90% confidence interval, we can use the method of estimating proportions. The confidence interval will give us a range within which the true proportion of successes (finding explosives) is likely to lie.

The formula for calculating the confidence interval is:
CI = p̂ ± Z * √[(p̂ * (1 - p̂)) / n],
where p̂ is the observed proportion of successes, Z is the Z-score corresponding to the desired confidence level (in this case, 90% confidence level), and n is the number of trials.

Let's calculate the p-value and the confidence interval using these formulas.

Calculating the p-value:
p-value = P(X ≥ 16), where X follows a binomial distribution with parameters n = 50 and p = 0.25.

Calculating the confidence interval:
CI = p̂ ± Z * √[(p̂ * (1 - p̂)) / n],

where p̂ = 16/50, Z = Z-score for 90% confidence level (approximately 1.645), and n = 50.

Let's calculate the p-value and the confidence interval:

p-value = P(X ≥ 16) = 1 - P(X < 16) ≈ 1 - 0.307,
p-value ≈ 0.693.

CI = (16/50) ± 1.645 * √[(16/50) * (1 - 16/50) / 50],
CI ≈ 0.32 ± 0.098,
CI ≈ (0.222, 0.418).

Based on the calculations, the p-value is approximately 0.693, which is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis that the device is not effective in sensing the presence of explosives.

The 90% confidence interval for the true proportion of successes (finding explosives) is approximately 0.222 to 0.418. This means that we can be 90% confident that the true proportion of times the device successfully detects explosives lies within this range.

In conclusion, based on the hypothesis test and the confidence interval, there is not enough evidence to suggest that the hand-held sensor is effective in sensing the presence of explosives.