1) Up-County Associates is hired to estimate the proportion of county households
ID: 3323307 • Letter: 1
Question
1) Up-County Associates is hired to estimate the proportion of county households that own three or more automobiles. They conducted their survey using a simple random sample of 250 randomly selected households and found that 23% of those households owned three or more automobiles. Answer the following questions, giving reasons for your answers. (a) Is the proportion, 23%, a parameter or a statistic? (b) If Up-County Associates were to continually choose simple random samples of 250 randomly selected households, each time recording the proportion of household in the sample owning three or more automobiles, what would be the “shape” of the resulting distribution of all these sample proportions? (c) Suppose now that survey company Lower-County Polling also conducted a survey to determine the proportion of county households that own three or more automobiles, but they surveyed only 100 randomly selected households. If Lower-County Polling were also to sample over and over again (using simple random samples of size 100) would the standard deviation of its distribution of sample proportions (i) in general, be lower than that of Up-County Associates? (ii) in general, be higher than that of Up-County Associates? (iii) in general, be about the same as that of Up-County Associates? (iv) in general, not have any predictable relationship to that of Up-County Polling? (d) Which company’s estimate is more likely to be closer to the actual proportion of households owning three or more automobiles?
Explanation / Answer
a) the proportion is the value of a statistic,not a parameter .since we only used a sample to estimate the value of the population parameter proportion and obtained 23 as the value
b) the shape of the proportion values can be bell-shaped. since we are taking a sufficiently large sample i.e. a sample of size 250, we may assume that the sample statistic (proportion) which is a binomial variate in this case, asymptotically becomes a normal variate, resulting in a bell-shaped curve.
c) if instead of 250, a sample of size 100 was used, then the variance (or, standard deviation) will increase as the standard error is inversely proportional to the sample size for a normally distributed variate.
d) up county associate will give a more precise estimate for the proportion of households owning 3 or more automobiles.as it has less standard error than the lower county
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