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. Please provide brief explanations for each.

The price of one gallon of gasoline at a particular gas station is recorded every day of the year, and the 365 values are plotted in a histogram.

Two hundred students in a statistics class each flip a coin 40 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 60 times and record the sum of the numbers they got on the 60 rolls. They make a histogram of the 200 sums.

One thousand randomly chosen people to report their annual salaries, and these salaries are plotted in a histogram.

The day before an election, fifty different polling organizations each sample 2000 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 2000 people in their sample, and the 50 averages are plotted in a histogram.

One hundred batteries are tested, and the lifetimes of the batteries are plotted in a histogram.

Answer 1

1. The price of one gallon of gasoline at a particular gas station is recorded every day of the year, and the 365 values are plotted in a histogram.

Gas prices generally have a tendency to increase over time and therefore the distribution is negatively skewed. Not bell curved.

2. Two hundred students in a statistics class each flip a coin 40 times and record the number of heads. The numbers of heads are plotted in a histogram.

This histogram will be bell curve because as the number of trials increases, the binomial distribution approaches normal.

3. Two hundred students in a statistics each roll a die 60 times and record the sum of the numbers they got on the 60 rolls. They make a histogram of the 200 sums.

This will also approach the normal distribution because as the number of rolls increases the distribution of the sums become symmetric.

4. One thousand randomly chosen people to report their annual salaries, and these salaries are plotted in a histogram.

Not normal. There is too much variation in people's annual salaries, especially if they belong to different professions. Since we chose the people randomly therefore this distribution would be highly skewed.

5. The day before an election, fifty different polling organizations each sample 2000 people and record the percentage who say they will vote for the Democratic candidate. The 50 values are plotted in a histogram.

This histogram will also approach normal distribution because voting follows binomial distribution and as the sample size increases, binomial distribution approaches normal.

6. The fifty polling organizations also record the average age of the 2000 people in their sample, and the 50 averages are plotted in a histogram.

Normal. Depending upon demographics of that particular area, the average age in the different samples should be close. Using the CLT, the average age should be approximately normal.

7. One hundred batteries are tested, and the lifetimes of the batteries are plotted in a histogram :

Here the sample size is large enough and as the sample size increases the distribution of the sample approaches to normal.

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