Question

In recent years more people have been working past the age of 65. In 2005, 27% of people aged 65–69 worked. A recent report from the Organization for Economic Co-operation and Development (OECD) claimed that percentage working had increased (USa today, November 16, 2012). The findings reported by the OECD were consistent with taking a sample of 600 people aged 65–69 and finding that 180 of them were working.
a. Develop a point estimate of the proportion of people aged 65–69 who are working.

b. Set up a hypothesis test so that the rejection of h0 will allow you to conclude that the

proportion of people aged 65–69 working has increased from 2005.

c. Conduct your hypothesis test using α 5 .05. What is your conclusion?

Answer 1

A point estimate of the proportion of people aged 65–69 who are working is p=0.3 or 30%, a hypothesis test so that the rejection of h0 will allow you to conclude that the  proportion of people aged 65–69 working has increased from 2005 is H_0: π =0.27 and H_1: π not equal to -0.27. My hypothesis test using α 5 .05 is that there is not enough evidence that the proportion of people aged 65–69 who are working has increased from 2005.

Explanation:

In recent years more people have been working past the age of 65. In 2005, 27% of people aged 65–69 worked. A recent report from the Organization for Economic Co-operation and Development (OECD) claimed that percentage working had increased (USa today, November 16, 2012). The findings reported by the OECD were consistent with taking a sample of 600 people aged 65–69 and finding

a. Develop a point estimate of the proportion of people aged 65–69 who are working.  By taking into account the sample that taken, the point estimate is   $p=\frac{180}{600} =0.3$

b. Set up a hypothesis test so that the rejection of h0 will allow you to conclude that the  proportion of people aged 65–69 working has increased from 2005.

In this case to claim that the proportion has changed and the mean is no longer 27%, we have to reject the null hypotesis (π=0.27). The null and alternative hypothesis are:

$H_0: \pi = 0.27\\ H_1 : \pi \neq 0.27$

c. Conduct your hypothesis test using α 5 .05. What is your conclusion?

Calculating the test statistic:

$\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.27(1-0.27)}{600} } = 0.01812$

The test statistic

$z=\frac{p-\pi-0.5/N}{\sigma} =\frac{0.3-0.27-0.5/600}{0.01812}=1.61$

It is a two-tailed test, P-value for z=1.61 is P=0.1074.

The hypothesis can not be rejected because as the P-value is bigger than the significance level, the effect is not statistically significant

Therefore there is not enough evidence that the proportion of people aged 65–69 who are working has increased from 2005.

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Answer 2

Answer:

a) point estimate is 30%

b) null and alternative hypothesis would be

$H_{0}$: p=27%

$H_{a}$: p>27%

c) We reject the null hypothesis, percentage working people aged 65-69 had increased

Step-by-step explanation:

a.

Point estimate would be the proportion of the working people aged 65–69 to the sample size and equals $\frac{180}{600}=0.3$ ie 30%

b.

Let p be the proportion of people aged 65–69 who is working. OECD claims that percentage working had increased. Then null and alternative hypothesis would be

$H_{0}$: p=27%

$H_{a}$: p>27%

c.

z-score of the sample proportion assuming null hypothesis is:

$\frac{p(s)-p}{\sqrt{\frac{p*(1-p)}{N} } }$ where

• p(s) is the sample proportion of working people aged 65–69 (0.3)
• p is the proportion assumed under null hypothesis. (0.27)
• N is the sample size (600)

then z=$\frac{0.3-0.27}{\sqrt{\frac{0.27*0.73}{600} } }$ = 1.655

Since one tailed p value of 1.655 = 0.048 < 0.05, sample proportion is significantly different than the proportion assumed in null hypothesis. Therefore we reject the null hypothesis.