Q.3 - The Practice of Statistics for AP

Question

Researchers in Norway analyzed data on the birth weights of $400,000$ newborns over a six-year period. The distribution of birth weights is approximately Normal with a mean of $3668$ grams and a standard deviation of $511$ grams.$9$ In this population, the range (maximum – minimum) of birth weights is $3417$ grams. We used Fathom software to take $500$ SRSs of size $n=5$ and calculate the range (maximum – minimum) for each sample. The dotplot below shows the results. 

(a) Is the sample range an unbiased estimator of the population range? Give evidence from the graph above to support your answer.

(b) Explain how we could decrease the variability of the sampling distribution of the sample range. 

Step-by-Step Solution

Answer

a) The population range may be estimated unbiasedly using the sample range.

b) By increasing the sample size, the variability in the distribution can be reduced.

Part(a) Step 1: Given information

Given in the question that, Researchers in Norway analyzed data on the birth weights of $400,000$ newborns over a six-year period. The distribution of birth weights is approximately Normal with a mean of $3668$ grams and a standard deviation of$511$grams$.9$ In this population, the range (maximum – minimum) of birth weights is $3417$ grams. We used Fathom software to take$500$ SRSs of size $n =5$ and calculate the range (maximum – minimum) for each sample. The dotplot below shows the results.

we need to find that whether the  sample range is an unbiased estimator of the population range.

Part(a) Step 2: Explanation

Given,

For the sample range to be an unbiased estimator of the population range, the range of the dot plot for every one of the samples should be same. However, here no sample is containing a similar range. Consequently, it's anything but an unbiased estimator of the population range.

Part(b) Step 1: Given information

Given in the question that, Researchers in Norway analyzed data on the birth weights of $400,000$ newborns over a six-year period. The distribution of birth weights is approximately Normal with a mean of $3668$ grams and a standard deviation of $511$ grams$.9$ In this population, the range (maximum – minimum) of birth weights is $3417$  grams. We used Fathom software to take 500 SRSs of size$n = 5$ and calculate the range (maximum – minimum) for each sample. The dotplot below shows the results.

We need to  Explain  that how we can decrease the variability of the sampling distribution of the sample range.

Part(b) Step 2: Explanation

The variability in the distribution could be diminished by expanding the sample size. Along these lines, to decrease the variability sample size should be expanded.