Assignment 3- Hypothesis Test for Proportions Instructions An issue that faces i
ID: 3047463 • Letter: A
Question
Assignment 3- Hypothesis Test for Proportions Instructions An issue that faces individuals investing for retirement is allocating assets among different investment choices. A study conducted 10 years ago showed that 65% of investors preferred stocks to real estate as an investment. In a recent random sample of 900 investors, 540 preferred stocks to real estate. Is this new data sufficient to allow you to conclude that the proportion of investors preferring stocks to real estate has declined from 10 years ago? Conduct your analysis at the 2% significance level. For help on this problem refer to Section 9.4 of the text on pages 406-9.Explanation / Answer
The following information is provided: The sample size is N = 900N=900, the number of favorable cases is X = 540X=540, and the sample proportion is ar p = rac{X}{N} = rac{ 540}{ 900} = 0.6p¯=NX=900540=0.6, and the significance level is lpha = 0.02=0.02
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho: p 0.65Ho:p0.65
Ha: p < 0.65Ha:p<0.65
This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.
(2) Rejection Region
Based on the information provided, the significance level is lpha = 0.02=0.02, and the critical value for a left-tailed test is z_c = -2.05zc=2.05.
The rejection region for this left-tailed test is R = {z: z < -2.05}R={z:z<2.05}
(3) Test Statistics
The z-statistic is computed as follows:
z = rac{ar p - p_0}{sqrt{p_0(1-p_0)/n}} = rac{ 0.6 - 0.65 }{sqrt{ 0.65(1- 0.65)/900}} = -3.145z=p0(1p0)/np¯p0=0.65(10.65)/9000.60.65=3.145
(4) Decision about the null hypothesis
Since it is observed that z = -3.145 < z_c = -2.05z=3.145<zc=2.05, it is then concluded that the null hypothesis is rejected.
Using the P-value approach: The p-value is p = 0.0008p=0.0008, and since p = 0.0008 < 0.02p=0.0008<0.02, it is concluded that the null hypothesis is rejected.
(5) Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion pp is less than p_0p0, at the lpha = 0.02=0.02 significance level.
Confidence Interval
The 98% confidence interval for pp is: 0.562 < p < 0.6380.562<p<0.638.