Question

For each of the questions below, a histogram is described. Indicate in each case whether, in view of the Central Limit Theorem, you can be confident that the histogram would look like approximately a bell-shaped (normal) curve, and give a brief explanation why (one sentence is probably sufficient). There are no data for these questions, so you will not need to use the computer to answer these questions.

a. A police department records the number of 911 calls made each day of the year, and the 365 values are plotted in a histogram.
b. The day before an election, fifty different polling organizations each sample 500 people and record the percentage who say they will vote for the Democratic candidate. The 50 values are plotted in a histogram. The fifty polling organizations also record the average age of the 500 people in their sample, and the 50 averages are plotted in a histogram.
c. One hundred batteries are tested, and the lifetimes of the batteries are plotted in a histogram.
d. Two hundred students in a statistics class each flip a coin 50 times and record the number of heads. The numbers of heads are plotted in a histogram. Two hundred students in a statistics each roll a die 40 times and record the sum of the numbers they got on the 40 rolls. They make a histogram of the 200 sums.
e. One thousand randomly chosen people report their annual salaries, and these salaries are plotted in a histogram.

Answer 1

Solution :

a). This histogram is not a bell shaped.

    Here we cannot use the theorem of the central limit as the histogram is being plotted by using 365 samples data values of the one sample.

b). This histogram may be bell shaped approximately since the sample size is large.

   Here there are 50 sample each having a size of 500 and there is 50 sample proportions. The histogram shows sampling distribution of the sample proportion. Thus the central limit theorem can be used.

The histogram is approximately bell shaped as there is 50 samples of each having 500 size and an average of 50. Thus the central limit theorem can be used.

c). The histogram is not a bell shaped.

   In this case central limit theorem cannot be used as the histogram is plotted by using 100 sample data values of one sample.

d). The histogram is a bell shaped since the sample size is large.

   In this case 200 samples  of each having a size of 50 and a data values of 50. The histogram here shows a sampling distribution of the sample proportion. Thus the central limit theorem can be used.

In the second also the histogram is of bell shaped as the sample size is large.

There are 200 samples of each sample having a size of 40 and they have 40 data values , i.e. sum of rolls.

e). It is not a bell shaped.

  The histogram here is plotted using 100 sample data values having one sample. Therefore we cannot use the central limit theorem.