Question

(22) The 99% confidence interval for the TRUE PROPORTION of success for a population is (0.318, 0.462). The random sample size is 300. (i) Please determine the SAMPLE proportion of success. (ii) Please determine the MARGIN FOR ERROR. (ii) Please determine the NUMBER OF SUCCESSFUL OUTCOMES. (23) The 90% confidence interval for the ACTUAL MEAN of a given population is (84, 90 ), via a "z" analysis. The random sample size is 81. (i) Please determine the (A) SAMPLE AVERAGE and (B) SAMPLE TOTAL (ii) Please determine the WIDTH of the confidence interval. (ii) Please determine the POPULATION VARIANCE. (24)You wish to carry out statistical inference regarding the population average. You wish the margin for error NOT to exceed 2, with 80% confidence. The population VARIANCE is 256. Please determine the SMALLEST ALLOWABLE SAMPLE SIZE for this experiment. (25) The hypothesized POPULATION VARIANCE 127. BUT, others think the data has LOWER DISPERSION than that. Please state (U) The NULL HYPOTHESIS symbolically: (i) the ALTERNATIVE hypothesis. (26)Ho : Po 0.37. However, more recent data now suggests that this original estimate of the population proportion may be underestimated. Please state the ALTERNATIVE hypothesis.

Answer 1

(22)

(i) Sample proportion of success ($\hat{p}$) is given by:

.4620.39 0.318+0.462

(ii)

Margin of Error (E) is given by:

E 0.462-0.39-0.072

(iii)

Number of successful outcomes (x) is given by:

r = n × p = 300 × 0.39= 117

(23)

(A)

Sample average ($\bar{x}$) is given by:

8490 7

(B)

Sample Total ($\sum x$) is given by:

87 × 81-7047

(ii)

Width (W) of confidence interval is given by:

(iii)

$W=Z\times SE$

For 0.10, Z = 1.645

Substituting:

3-1.645 × SE

So

1.8237 SE 1.645

$SE=\frac{\sigma }{\sqrt{n}}=1.8237$

Substituting n = 81, we get:

$\sigma =1.8237\times 9=16.4134$

(24)

$n=\frac{Z^{2}_{\alpha /2}\times \sigma ^{2}}{e^{2}}$

For $\alpha$ = 0.20, Z = 1.28

$\sigma =\sqrt{256}=16$

e = 2

Substituting, we get:

$n=\frac{1.28^{2}\times 16^{2}}{2^{2}}=105$

(25)

(i)

H0: Null Hypothesis: $\sigma ^{2}$ = 127

(ii)

HA: Alternative Hypothesis: $\sigma ^{2}$ < 127

(26)

HA: Alternative Hypothesis: $\sigma ^{2}$ > 127