Question

Explain hypothesis testing to a friend, using the following scenario as a model. Describe the hypotheses, the sample statistic, the P-value, the meanings of Type I and Type II errors, and the level of significance. Discuss the significance of the results. Formulas are not required.

A team of research doctors designed a new knee surgery technique utilizing much smaller incisions than the standard method. They believe recovery times are shorter when the new method is used. Under the old method, the average recovery time for full use of the knee is μ₁ = 4.5 months. A random sample of 36 surgeries using the new method showed the average recovery time to be μ₂ = 3.7 months, with sample standard deviation of 1.7 months. The P-value for the test is 0.0039. The research team states that the results are statistically significant at the 1% level of significance.

Describe the hypotheses.

Null hypothesis: the average recovery times are the same; Alternate hypothesis: the average recovery time for the new method is less than the average recovery time for the old method

Null hypothesis: the average recovery times are the same; Alternate hypothesis: the average recovery time for the new method is not the same as the average recovery time for the old method    

Null hypothesis: the average recovery times are the same; Alternate hypothesis: the average recovery time for the new method is greater than the average recovery time for the old method

Null hypothesis: the average recovery time for the new method is less than the average recovery time for the old method; Alternate hypothesis: the average recovery times are the same

What does the P-value mean?

It is the probability that a sample could be gathered from the population with the given characteristics.

It is the probability that the population could have a sample like this again.    

It is the probability that the sample has that mean.

It is the probability that the population actually has that mean.

What is the meaning of a Type I error?

Accept the hypothesis that the average recovery times are the same when in fact this is true.

Accept the hypothesis that the average recovery times are the same when in fact this is false.    

Reject the hypothesis that the average recovery times are different when in fact this is true.

Reject the hypothesis that the average recovery times are different when in fact this is false.

What is the meaning of a Type II error?

Accept the hypothesis that the average recovery times are the same when in fact this is true.

Accept the hypothesis that the average recovery times are the same when in fact this is false.    

Reject the hypothesis that the average recovery times are different when in fact this is true.

Reject the hypothesis that the average recovery times are different when in fact this is false.

What does it mean to be statistically significant at the 1% level of significance?

That the P-value is less than the 1% level that we are testing it against, allowing us to reject the null hypothesis.

That the P-value is greater than the 1% level that we are testing it against, allowing us to fail to reject the null hypothesis.    

That the P-value is less than the 1% level that we are testing it against, allowing us to accept the alternate hypothesis.

That the P-value is greater than the 1% level that we are testing it against, allowing us to reject the null hypothesis.

That the P-value is less than the 1% level that we are testing it against, allowing us to fail to reject the null hypothesis.

Answer 1

Describe the hypotheses.

a) Null hypothesis: the average recovery times are the same; Alternate hypothesis: the average recovery time for the new method is less than the average recovery time for the old method

Since, the objective is to determine if the new method is better

What does the P-value mean?

d) It is the probability that the population actually has that mean.

Since, under null hypothesis how probable it is get that mean value

What is the meaning of a Type I error?

c) Reject the hypothesis that the average recovery times are different when in fact this is true.

Reject the null hypothesis when it is true ( mean are same )

What is the meaning of a Type II error?

b) Accept the hypothesis that the average recovery times are the same when in fact this is false.   

Accept the alternate hypothesis when it is false

What does it mean to be statistically significant at the 1% level of significance?

a) That the P-value is less than the 1% level that we are testing it against, allowing us to reject the null hypothesis.

We reject H0 when p is less than alpha