Question

a. Your boss asks you to conduct a hypothesis test about the mean dwell time of a new type of UAV. Before you arrived, an experiment was conducted on n = 5 UAVs (all of the new type) resulting in a sample mean dwell time of ybar = 10.4 hours. The goal is to conclusively demonstrate, if possible, that the data supports the manufacturerer's claim that the mean dwell time is greater than 10 hours. Given that is is reasonable to assume the dwell time are normally distributed, the sample standard deviation is s = .5 hours, and using a significance level of alpha = .01, conduct the appropriate hypothesis test.

b. For the hypothesis test constructed, what is the probability of a Type II error if the true mean is 10.4 hours? Interpret in your own words.

c. What is the probability of a Type II error if the true mean were actually 11 hours? What do your answers from b. and c. tell you about the power of your hypothesis test?

Answer 1

a.

Null Hypothesis(H0):

The mean dwell time is not significantly greater than 10 hours. 10.

(Hypothesised population mean, =10)

Alternative Hypothesis(H1):

The mean dwell time is significantly greater than 10 hours. > 10. (right-tailed test).

(where, =Population mean dwell time).

Test statistic:

Since the sample size, n =5 < 30 (small sample), we shall use t-score.

Standard Error, SE =s/ =0.5/ =0.2236

Test statistic, t =(​​​​​​)/SE =(10.4 - 10)/0.2236 =1.789

Critical value:

At 0.01 significance level, at df =n-1 =4, for a right-tailed test, t-critical =3.747

Decision criteria:

Since it is a right-tailed test, reject H0 if t > t-critical

Conclusion:

Since t: 1.789 < t-critical: 3.747, we failed to reject the null hypothesis (H0) at 1% significance level.

Thus, we do not have sufficient statistical evidence to claim that the mean dwell time is greater than 10 hours.

b.

Type II error means 'failing to reject the null hypothesis when it is false'.

We need to calculate the probability of a Type II error conditional on a particular value of µ.

Given that µ = 10.4 hours

Since it is a right-tailed test, we will fail to reject the null hypothesis (commit a Type II error) if we get a test statistic(t) less than 3.747

t-critical =(​​​​​​)/SE

3.747 =()/0.2236 =(3.747*0.2236)+10 =10.84

So, we will incorrectly fail to reject the null hypothesis as long as we draw a sample mean that is less than 10.84. The probability of drawing a sample mean less than 10.84 given = 10.4 and SE =0.2236 is:

P(​​​​​​ < 10.84) =P[Z < (10.84 - 10.4)/0.2236] =P(Z < 1.9678) =0.9755

Thus, the probability of a Type II error =0.9755

Interpretation:

"The probability of not rejecting the null hypothesis when true mean is 10.4 is 0.9755".

c.

Given that =11 hours

The probability of drawing a sample mean less than 10.84 given =11 and SE =0.2236 is:

P( < 10.84) =P[Z < (10.84 - 11)/0.2236] =P(Z < −0.7156) =0.2371

Thus, the probability of a Type II error =0.2371

Interpretation:

"The probability of not rejecting the null hypothesis when true mean is 11 is 0.2371".

Power of hypothesis test:

Power is the probability of rejecting the false null hypothesis.

Power =1 - P(type II error)

For b. Power =1 - 0.9755 =0.0245

For c. Power =1 - 0.2371 =0.7629

This means, the farther the true mean is from the null (i.e., hypothesised mean), the more the power of the test will be".