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Question: For each of the questions below, a histogram is described. Indicate in each case whether, in view...

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).

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.

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.

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.

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

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.

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.

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

Answer 1

The Central Limit Theorem(CLT) states that - "given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a finite expected value and a finite variance, will be approximately normally distributed, regardless of the original distribution of the population".

I can be confident that the histogram would look like approximately a bell-shaped (normal) curve for 2 and 3 cases because the Central Limit Theorem is applicable for them and the sample sizes are large enough (200 and 200).

Fot the cases of 1, 4, 5, 6 and 7 the Central Limit Theorem cannot be applied and thus, I cannot be confident that the histogram would look like approximately a bell-shaped (normal) curve. For 5 and 6 cases, the polling organizations sample the same population members and so, they are not independent. For 1, 4 and 7 cases, they are the original populations and I cannot comment on the distribution as they may be any kind of distribution, normal or other.