Question

A hospital reported that the normal death rate for patients with extensive burns (more than 40% of skin area) has been significantly reduced by the use of new fluid plasma compresses. Before the new treatment, the mortality rate for extensive burn patients was about 60%. Using the new compresses, the hospital found that only 42 of 89 patients with extensive burns died. Use a 1% level of significance to test the claim that the mortality rate has dropped.

What are we testing in this problem?

single proportionsingle mean     

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ = 0.6; H₁:  μ < 0.6

H₀: μ = 0.6; H₁: μ > 0.6     

H₀: μ = 0.6; H₁:  μ ≠ 0.6

H₀: p = 0.6; H₁:  p ≠ 0.6

H₀: p = 0.6; H₁: p > 0.6

H₀: p = 0.6; H₁:  p < 0.6

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal, since np > 5 and nq > 5.

The standard normal, since np < 5 and nq < 5.     

The Student's t, since np < 5 and nq < 5.

The Student's t, since np > 5 and nq > 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.250

0.125 < P-value < 0.250     

0.050 < P-value < 0.125

0.025 < P-value < 0.050

0.005 < P-value < 0.025

P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.     

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.01 level to conclude that the mortality rate has dropped.

There is insufficient evidence at the 0.01 level to conclude that the mortality rate has dropped.

Answer 1

Solution :

Given that,

n = 89

x = 42

= x / n = 42/89 = 0.47

(a) The level of significance is 0.01

The null and alternate hypotheses

H0: p = 0.6

H1:  p < 0.6

(b) The standard normal, since np > 5 and nq > 5.

Test Statistic:

Z =   - P / ( ((P * (1 - P)) / n)

Z = 0.47- 0.6/  ( ((0.6* 0.4) / 89)

Z =

Z = -2.50

(c) P- value = 0.0062

0.005 < P-value < 0.025

(d) At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) There is sufficient evidence at the 0.01 level to conclude that the mortality rate has dropped.