Question

In a previous​ year, 51​% of females aged 15 and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 550 females aged 15 and older and finds that 287 are living alone. Is there sufficient evidence at the alphaequals0.01 level of significance to conclude the proportion has​ changed? Because np 0 (1 - p 0 ) equals nothing______not equals greater than less than equals ​10, the sample size is______greater than less than ​5% of the population​ size, and the sample______cannot be reasonably assumed to be random, can be reasonably assumed to be random, is given to be random, is given to not be random, the requirements for testing the hypothesis______are not are satisfied.
1. Find the test statistic for this hypothesis test. z 0 equals nothing.
2. Determine the​ P-value for this hypothesis test. ​P-value equals nothing.
3. State the conclusion for this hypothesis test.
A. Do not reject Upper H0. There is not sufficient evidence at the alpha equals 0.01 level of significance to conclude that the proportion of females who are living alone has changed.
B. Do not reject Upper H0. There is sufficient evidence at the alphae quals 0.01 level of significance to conclude that the proportion of females who are living alone has changed.
C. Reject Upper H0. There is sufficient evidence at the alpha equals 0.01 level of significance to conclude that the proportion of females who are living alone has changed.
D. Reject Upper H0. There is not sufficient evidence at the alpha equals 0.01 level of significance to conclude that the proportion of females who are living alone has changed.

Answer 1

Answer:

Given:

n = 550

Po = 51% = 0.51

P0 - 1 = 0.51 - 1 = 0.49

p' = $\frac{287}{550}$ = 0.5218

np0(1 - p0) = 550*0.51(1-0.51)

= 137.4 ≥ 100

This means the sample size is greater than 5% of the population size. and the sample can be reasonably assumed to be random, the requirement for testing hypothesis are satisfied.

1) The null and alternative hypotheses:

H0 : p0 = 0.51

H1 : p0 ≠ 0.51

The test statistic Z,

$Z = \frac{p' - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

$Z = \frac{0.5218 - 0.51}{\sqrt{\frac{0.51(1-0.51)}{550}}}$

Z = 0.55

This is two tailed test.

The pvalue for Z =0.55

Pvalue = 0.5824

3) Decision:

Since pvalue, 0.5824 is greater than significance level, we fail to reject null hypothesis H0.

Conclusion:

Do not reject Upper H0. There is not sufficient evidence at the alpha equals 0.01 level of significance to conclude that the proportion of females who are living alone has changed.