Law of Averages
13. People in a large population average 60 inches tall. You will take a random sample and will be given a dollar for each person in your sample who is over 65 inches tall. For example if you sample 100 people and 20 turn out to be over 65 inches tall, you get $20. Which is better: a sample of size 100 or a sample of size 1,000? Choose one and explain. Does the law of averages relate to the answer you give?
In this case a sample size of 100 would be better. This can be explained using law of averages and also by looking at the formula for Margin of Error.
ME = 2 * SD of population/SQRT(size of sample)
In this case, the SD of population would remain the same. The only factor or value changing is the sample size. Sample size of 100 would give a bigger ME value as compared to sample size = 1000. With higher value of ME the possibility of values for height over 65 is more. However, it does not completely confirm that more values would be over 65 for sure.
Also looking at the law of averages for a sample mean, “with a large randomly selected sample, the sample average tends to be close to the true population average”. In this case it is 60 inches. “The larger the sample, closer is sample average to population average”. So if we take sample size of 1000, the average of sample would be closer to 60 inches.
Here we are assuming that the sample size of 100 and 1000 is selected without any kind of bias.
18. Based on a simple random sample of one hundred an analyst estimates the average hourly wage earned by workers in a city to be $30 and computes the margin of error to be $5. Can we conclude from this that most workers there earn between $25 and $35 an hour? Is this the right interpretation for the margin of error?
Margin of error as per the text is “it estimates the largest distance you would reasonably expect to see between sample average and population average”1. Based on the data provided above, with ME = $5 we can be confident that in the population from which
In this case a sample size of 100 would be better. This can be explained using law of averages and also by looking at the formula for Margin of Error.
ME = 2 * SD of population/SQRT(size of sample)
In this case, the SD of population would remain the same. The only factor or value changing is the sample size. Sample size of 100 would give a bigger ME value as compared to sample size = 1000. With higher value of ME the possibility of values for height over 65 is more. However, it does not completely confirm that more values would be over 65 for sure.
Also looking at the law of averages for a sample mean, “with a large randomly selected sample, the sample average tends to be close to the true population average”. In this case it is 60 inches. “The larger the sample, closer is sample average to population average”. So if we take sample size of 1000, the average of sample would be closer to 60 inches.
Here we are assuming that the sample size of 100 and 1000 is selected without any kind of bias.
18. Based on a simple random sample of one hundred an analyst estimates the average hourly wage earned by workers in a city to be $30 and computes the margin of error to be $5. Can we conclude from this that most workers there earn between $25 and $35 an hour? Is this the right interpretation for the margin of error?
Margin of error as per the text is “it estimates the largest distance you would reasonably expect to see between sample average and population average”1. Based on the data provided above, with ME = $5 we can be confident that in the population from which