统计代写|MTH432 Sampling theory

Stratified $R S$ with Optimal allocation.
Based on strata population sizes and sample variances for each strata, we calculate optimal sample sizes and round for $n_{\mathrm{opt}}=(28,51,46,38,37)$. We note the sample sizes for the middle strata are similar to the proportional allocation, 8 less in the first strata, and 6 more in the last strata. The table below gives the estimates (sample mean and proportion) with a $95 \% \mathrm{CI}$, and the standard error of the estimate.

For Price, the SE for the optimal allocation differs by 0.0024 , which is not effectively different than our proportional allocation. The $95 \%$ is wider by only a penny for price. For proportion $\leq 11$ yo., the SE differs by only 0.0004 , a negligible difference.
\begin{tabular}{crcr}
& Est & $95 \%$ CI & SE(Est) \
\hline Price & 9.9653 & $(9.2597,10.6710)$ & 0.3600 \
$\leq 11$ yo. & 0.3901 & $(0.3245,0.4558)$ & 0.0335 \
\hline
\end{tabular}

Assignment 3.2.5 Optimal vs. proportional allocation.
Optimal allocation is expected to perform better than proportional allocation when the variance within strata differ.

Based on our stratification, this did not occur in Lockhart City. Optimal allocation did not give different standard errors from proportional allocation because each of our strata have similar variances (see 3.2.6).

Deficiencies in stratification.
There does not seem to be any deficiencies in our stratification for the purposes of reducing variance in the estimate of price willing to pay for cable. Levene’s test for unequal variances shows that our strata variances are not different from each other (see “Price willing to pay for cable” results below). That indicates that we chose strata that are self similar in the price willing to pay for cable. The ANOVA below strongly indicates differences between strata ( $\mathrm{p}$-value $<0.001$ ) (see the results below).

There are deficiencies in our stratification for the purposes of reducing variance in the estimate of proportion of children aged 11 and younger. This is because our stratification variable of average house value is not related to whether the household has children. This could be improved by also stratifying on the average number of people per household, with larger numbers being more likely to have children aged 11 and younger. Levene’s test for unequal variances shows that our strata variances are not different from each other (we understand that this test does not apply since the data do not come from a continuous distribution) (see “Proportion of households with children 11 years and younger” results below). That indicates that we chose strata that are self similar in the proportion of households with children aged 11 and younger. The ANOVA below indicates no difference between strata (p-value $=0.333)$ (see the results below).

Negligible differences in SE in optimal allocation versus proportional allocation is attributed to our strata having similar variances. However, if we observed a stratum that had a much larger variance than the other strata, then observations in that strata are less self-similar than in other strata, indicating that the stratification scheme may need revision.

In cluster sampling, it is desired that each cluster be representative of the population as a whole.
(a) Use stratified rather than cluster sampling since blocks are selfsimilar, but different from one another.
(b) Cluster sampling may be reasonable if the constant proportion of nonwhites in each block is near the population proportion of nonwhites. That is, cluster sampling would be ideal if the population of interest is entirely within the sampling frame of blocks.
(c) A situation for cluster sampling, each cluster is as a SRS from the population.

(a) This is a one-stage cluster sample with PSU=scholarly journals in the social and behavioral sciences, and SSU=articles published during 1988 from the selected journals in the PSU.
(b) Estimate.
$$
\begin{aligned}
\hat{p}r & =\frac{\sum{i=1}^n t_i}{\sum_{i=1}^n M_i}=\frac{137}{148}=0.9257 \
\mathrm{SE}\left(\hat{p}_r\right) & =\sqrt{\operatorname{Var}\left(\hat{p}_r\right)}=\sqrt{\frac{\hat{p}_r\left(1-\hat{p}_r\right)}{n}}=\sqrt{\frac{0.9257(1-0.9257)}{26}}=0.0514
\end{aligned}
$$
Giving a $95 \%$ CI for $p$ of $(0.8249,1)$ (upper CI limit of 1.0265 truncated at 1).

(c) Ridiculous reasoning.
Social and behavioral sciences are jumping from a bridge… should our courts of law? The purpose of probability sampling is to get an unbiased picture of the population of interest. An nonprobability sample can make no guarantee about the accuracy of the estimates obtained, since the issue of bias has not been addressed.

Consider a case involving trademark violation or employment discrimination. A nonprobability sample might well include a convienience sample which includes a much higher-than-average proportion of violations or cases of discrimination, or a much lower-than-average proportion (depending whether the prosecution or defense is making the case). Since we SHOULD be interested in making an honest and truthful case, a probability sample done correctly is the only likely way to obtain an accurate estimate of the population. Without truth, we may as well let belief blindly lead.