Solved – How to make sample space of the following question

samplesample-sizesamplingself-study

My teacher gave me a question to solve.In the book of Ronald E. Walpole (Probability and Statistics for Engineers and Scientists), chapter no#3 exercise 1,question #2 which is as below:

3.2) An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 3 of these
automobiles at random, list the elements of the sample space S, using
the letters B and N for blemished and nonblemished, respectively; then
to each sample point assign a value x of the random variable X
representing the number of automobiles with paint blemishes purchased
by the agency.

It means we have 2 DEFECTIVE AND 3 NON-DEFECTIVE automobiles;

But in the solution manual he makes samples as follows:

Sample Space 
NNN 
NNB 
NBN 
BNN 
NBB 
BNB 
BBN 
BBB

but the question is how did he make the last sample {BBB}.I think it is not possible because we have maximum 2 blemished automobiles.and we are recieving 3 means repetition not allowed.Please tell me how {BBB} is possible.

Best Answer

BBB would have been a possible sample if the simple random sample had been drawn with replacement. In with-replacement simple random sampling, each selected item is "replaced" before the next draw is made. Each of the three draws is made from the original five, with probability of selection 1 in 5 at each draw.

This is a valid sampling method, but the description of the problem makes it clear that sampling was without replacement. I conclude that you are correct: A BBB sample point is impossible, and the listing in the solutions manual is in error.

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Solved – How to make representative sample set from a large overall dataset

If you don't wish to parse the entire data set then you probably can't use stratified sampling, so I'd suggest taking a large simple random sample. By taking a random sample, you ensure that the sample will, on average, be representative of the entire dataset, and standard statistical measures of precision such as standard errors and confidence intervals will tell you how far off the population values your sample estimates are likely to be, so there's no real need to validate that a sample is representative of the population unless you have some concerns that is was truly sampled at random.

How large a simple random sample? Well, the larger the sample, the more precise your estimates will be. As you already have the data, conventional sample size calculations aren't really applicable -- you may as well use as much of your dataset as is practical for computing. Unless you're planning to do some complex analyses that will make computation time an issue, a simple approach would be to make the simple random sample as large as can be analysed on your PC without leading to paging or other memory issues. One rule of thumb to limit the size of your dataset to no more than half your computer's RAM so as to have space to manipulate it and leave space for the OS and maybe a couple of other smaller applications (such as an editor and a web browser). Another limitation is that 32-bit Windows operating systems won't allow the address space for any single application to be larger than $2^{31}$ bytes = 2.1GB, so if you're using 32-bit Windows, 1GB may be a reasonable limit on the size of a dataset.

It's then a matter of some simple arithmetic to calculate how many observations you can sample given how many variables you have for each observation and how many bytes each variable takes up.

Solved – Sample size of sampling distribution of the sample mean

Yes, it is true that the sampling distribution of the mean is equal to the population mean regardless of how small or large the sample sizes are (In other words, the mean is ubiased).

A good way to think about this is to take a small population and study it. Say for example your population consists of 4 elements having values 2, 4, 6, and 10. The sampling distribution can be thought of as taking samples of a certain size over and over again from this population. So for example you wanted to sample 3 elements. Then, on average, the mean of the means of a lot of samples of size 3 will be unbiased (e.g. the sample means of the mean will be a good approximation of the true population mean). To see this note that the population average of 2, 4, 6, and 10 is 5.5. The following R code shows 100000 samples of size 3 and of size 2 and of size 1. After each sample is selected, the mean is calculated. After all 100000 sample means are calculated, we find the sample mean of those means. Note that this mean is very close to the population mean of 5.5 in every case (the estimates round to 5.5 every time), regardless of how large the sample size is:

x<-c(2,4,6,10)
> #Population Mean
> mean(x)
[1] 5.5
> 
> sample.means<-rep(NA, 100000)
> 
> set.seed(5)
> #Sample of size n=3
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=3))
+ }
> mean(sample.means)
[1] 5.4982
> 
> #Sample of size n=2
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=2))
+ }
> mean(sample.means)
[1] 5.49388
> 
> #Sample of size n=1
> 
> for (i in 1:100000){
+   sample.means[i]<-mean(sample(x, size=2))
+ }
> mean(sample.means)
[1] 5.49419

Best Answer

Related Solutions

Solved – How to make representative sample set from a large overall dataset

Solved – Sample size of sampling distribution of the sample mean

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