Question

Problem: Proportion of "Cured" Cancer Patients: How Does Canada Compare with Europe?

Lung cancer remains the leading cause of cancer death for both Canadian men and women, responsible for the

most potential years of life lost to cancer. Lung cancer alone accounts for 28% of all cancer deaths in Canada

(32%. in Quebec). Most forms of lung cancer start insidiously and produce no apparent symptoms until they

are too far advanced. Consequently, the chances of being cured of lung cancer are not very promising, with the

ve-year survival rate being less than 15%. The overall data for Europe show that the number of patients who

are considered "cured" is rising steadily. For lung cancer, this proportion rose from 6% to 8%. However, there

was a wide variation in the proportion of patients cured in individual European countries. For instance, the study

shows that for lung cancer, less than 5% of patients were cured in Denmark, the Czech Republic, and Poland,

whereas more than 10% of patients were cured in Spain.

1. Suppose a sample of 75 Quebecers was selected and it was found that 27 of them had died due to lung

cancer.

(a) Perform the appropriate test of hypothesis to determine whether the proportion of deaths in Quebec

due to the leading cause (lung cancer) has changed. Test using = 0:05.

(b) Find the p-value for this test and interpret it.

(c) Construct a 95% condence interval estimate of the population proportion of the premature cancer

death in Quebec.

(d) Explain how to use this condence interval to test the hypotheses.

2. Suppose two independent samples were taken. The following data were recorded:

Quebec: n1 = 150 Number of deaths due to cancer= x1 = 47

Rest of Canada: n2 = 1000 Number of deaths due to cancer= x2 = 291

(a) Suppose the scientists have no preconceived theory concerning which proportion parameter is the larger

and they wish to detect only a dierence between the two parameters, if it exists. What should they

choose as the null and alternative hypotheses for a statistical test?

(b) What type of error could occur in testing the null hypothesis in (a), if H0 is false?

(c) Calculate the standard error of dierence between the two sample proportions. Make sure to use the

pooled estimate for the common value of true proportion.

(d) Calculate the test statistic that you would use for the test in (a). Based on your knowledge of the

standard normal distribution, is this a likely or unlikely observation, assuming that H0 is true and the

two population proportions are the same?

(e) Find the p-value for the test. Test for a signicant dierence between the population proportions at

the 1% signicance level.

(f) Find the rejection region when = 0:01. Using critical value approach, determine whether the data

provide sucient evidence to indicate a dierence between the population proportions.

(g) Use a 95% condence interval to estimate the actual dierence between the cancer death proportions

for the people in Quebec versus rest of the Canada. Summarize your ndings.

3. To test the claim regarding proportion of cured cancer patients in Spain, a random sample of 500 patients

yielded 39 who were cured.

(a) State the appropriate null and alternative hypotheses.

(b) Calculate the value of the test statistic.

(c) Calculate the p-value and write your conclusion.

Answer 1

32%. in Quebec

1. Suppose a sample of 75 Quebecers was selected and it was found that 27 of them had died due to lung

cancer.

(a) Perform the appropriate test of hypothesis to determine whether the proportion of deaths in Quebec

due to the leading cause (lung cancer) has changed. Test using = 0:05.

It is given that

Null and alternate hypothesis:

Level of significance:

Test statistic: will have standard normal distribution.

  

(b) Find the p-value for this test and interpret it.

The p-value of the test is 0.4577 obtained through 2*(1-NORM.S.DIST(0.7426,TRUE)) function in EXCEL.

(c) Construct a 95% confidence interval estimate of the population proportion of the premature cancer

death in Quebec.

The 95% confidence interval is given by

  

  

  

  

(d) Explain how to use this confidence interval to test the hypotheses.

We need to check whether the null value ie 0.32 is contained in this interval or not. If this interval contains the null value, then the Null hypothesis is not rejected. Since this interval contains the Null value, we fail to reject the Null Hypothesis.

2. Suppose two independent samples were taken. The following data were recorded:

Quebec: n1 = 150 Number of deaths due to cancer= x1 = 47

Rest of Canada: n2 = 1000 Number of deaths due to cancer= x2 = 291

The pooled proportion

(a) Suppose the scientists have no preconceived theory concerning which proportion parameter is the larger

and they wish to detect only a difference between the two parameters, if it exists. What should they

choose as the null and alternative hypotheses for a statistical test?

(b) What type of error could occur in testing the null hypothesis in (a), if H0 is false?

Type II error

(c) Calculate the standard error of difference between the two sample proportions. Make sure to use the

pooled estimate for the common value of true proportion.

The standard error is given by

  

  

  

  

(d) Calculate the test statistic that you would use for the test in (a). Based on your knowledge of the

standard normal distribution, is this a likely or unlikely observation, assuming that H0 is true and the

two population proportions are the same?

The test statistic is

  

.

Since in a standard Normal distribution, this value is quite likely and the Null hypothesis will not be rejected.(The citical value is 1.96).

(e) Find the p-value for the test. Test for a significant difference between the population proportions at

the 1% significance level.

The p-value is 0.5755

(f) Find the rejection region when = 0:01. Using critical value approach, determine whether the data

provide sufficient evidence to indicate a difference between the population proportions.

The critical region is

Since our calculated value of Z=0.5591, doesn't fall in the region of rejection, we fail to reject the null hypotheis.

(g) Use a 95% confidence interval to estimate the actual difference between the cancer death proportions

for the people in Quebec versus rest of the Canada. Summarize your findings.

The 95% confidence interval for the difference in proportions is given by:

The 95% confidence interval for the difference in the proportions is (-0.0570,0.1016). Since this interval contains 0, we fail to reject the Null hypothesis.

3. To test the claim regarding proportion of cured cancer patients in Spain, a random sample of 500 patients

yielded 39 who were cured.

Here we have

(a) State the appropriate null and alternative hypotheses.

Since we are given that more than 10% of patients were cured in Spain.

(b) Calculate the value of the test statistic.

(c) Calculate the p-value and write your conclusion.

The p-value of the test is 0.9495. Since the p-value>0.05, we fail to reject the Null hypothesis and conclude that there is no evidence in the claim that the proportion cured is more than 10%.