Question

The director of research and development is testing a new medicine. She wants to know if there is evidence at the 0.01 level that the medicine relieves pain in more than 361 seconds. For a sample of 62 patients, the mean time in which the medicine relieved pain was 366 seconds. Assume the population standard deviation is 20. Make the decision to reject or fail to reject the null hypothesis.

and

The mayor of a town has proposed a plan for the construction of a new community. A political study took a sample of 900 voters in the town and found that 81% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 78%. Testing at the 0.02 level, is there enough evidence to support the strategist's claim?

There is sufficient evidence to support the claim that the percentage of residents who favor construction is more than 78%.

There is not sufficient evidence to support the claim that the percentage of residents who favor construction is more than 78%.

Answer 1

1)

Answer)

As the population standard deviation is known here we can use standard normal z table to estimate the answer

Ho : u = 361

Ha : u > 361

Given significance level is 0.01

From z table, p(z>2.33) = 0.01

So, rejection rule is if calculated z is > 2.33, reject null hypothesis

Z = (sample mean - claimed mean)/(s.d/√n)

Z = (366-361)/(20/√62)

Z = 1.97

As it is not greater than 2.33

So, we fail to reject the null hypothesis

Second)

Ho : P = 0.78

Ha : P > 0.78

Test statistics Z = (observed p - claimed p)/standard error

Standard error = √claimed p*(1-claimed p)/√n

N = 900

Claimes p = 0.78

Observed p = 0.81

Z = 2.17

From z table, p(z>2.05) = 0.02

So, rejection region is if z >2.02

Reject Ho

As 2.17 is greater than 2.02

We reject Ho

And

There is sufficient evidence to support the claim that the percentage of residents who favor construction is more than 78%.