Question

The following is a list of sample means for human adult height, in cm. Each was calculated in the same way from samples taken from the same hypothetical population.

160.5, 162.5, 161.7, 160.2, 163.7, 159.8, 160.6, 161.6

The true mean of the population is 158.7 cm. Use the jargon of estimation to describe the likely type of problem in the sampling process.

Answer 1

Here we have the list of sample means for human adult height. The heights are:

160.5,162.5,161.7,160.2,163.7,159.8,160.6,161.6 (in cm)

The true population mean is 158.7 cm

Now we can talk about some measures of central tendency from the given data.

We know, if x_i's are the observations, then the mean of the data we be

where n is the total number of observations.

here for this problem, n= 8 and mean of the sample means is =

=

=161.325

Also we know that, median of a dataset is the middlemost value of the data set after arranging the whole data in ascending or descending order. Here, in this case if we arrange the whole data in ascending order then we get,

159.8,160.2,160.5,160.6,161.6,161.7,162.5,163.8

Here we have even number (8) of observations, hence we know the median will be the mean of (8/2)^th and ((8/2)+1)^th observation, that is the mean of 4th and 5th observation of the ordered data set.

Here mean of the 4th and 5th data point is (160.6+161.6)/2 = 161.2 (in cm)

We know the true population mean is 158.7 cm.

But here the sample population mean is 161.325 cm

We know that sample mean is the unbiased estimator of the population mean.

But here in this case , we can observe that each of the 8 sample means are greater than the true population mean.

Hence we can conclude that either we are not taking a proper sample, that is , sample is not the good representative of the population, or we can formulate a hypothesis testing of mean, where the null hypothesis H₀ : true mean = 158.7 against H₁ : true mean > 158.7 cm

Assuming normal population we can proceed the testing, which will lead to a t-test because population mean is not known, hence we have to estimate it using the sample variance S² (with the divisor n-1) , where sample mean and sample variance are independently distributed.

If we can get enough evidence at some % level of significance then we can conclude that the true population mean is greater than 158.7 cm at that % of level of significance.

Here the mean of the sample means is quite large than the true population mean. So, based on this sample, we can intuitively conclude that the null hypothesis will be rejected at some level of significance .

Also we know that sample mean is consistent estimator of population mean.

The above statement means that if we increase the number of samples then it will eventually tend to the true population mean.

Here the sample size is 8 . If we increase the sample size then we can have the true population mean based on the consistency concept of sample mean.

Here sample median is also greater than true population mean. This data clearly indicates that the true population mean should be greater than 158.7 cm.