3. Suppose 20% of all heart transplant patients do not survive the operation.
a. Think about taking repeated random samples of 371 patients from this population.
Describe how the sample proportion who die would vary from sample to sample. (Hint: Be sure to refer to the shape, center, and spread of its sampling distribution.) Also include a well-labeled sketch to represent this distribution.
b. Suppose you take a random sample of 371 heart transplant patients. Determine the probability that the sample proportion who die would be .213 or higher.
4. Suppose I tell you that I flipped a coin multiple times and got 75% heads. Would you be reasonably convinced that this was not a fair coin (where “fair” means that the coin has a 50/50 chance of landing heads or tails)? If so, explain why. If not, describe what additional information you would ask for and explain why that information is necessary.
5. The distribution of house prices is skewed to the right because most houses cost a modest amount but a few cost a very large amount. If you take a random sample of 1000 houses, can you reasonably expect the distribution of the house prices to be approximately normal? Explain your answer.
6. Suppose you flip a fair coin until it lands heads up for the first time. It can be shown (do not try to calculate this) that the expected value of the number of flips required is 2. Explain (with a sentence or two) what this expected value means in this context.
3. When taking repeated random samples of 371 patients from a population where 20% do not survive the operation, the sample proportion who die would vary from sample to sample. The shape of the sampling distribution would be approximately bell-shaped or normal. The center of the distribution or the mean of the sample proportion who die would be the same as the population proportion, which is 20%. The spread of the distribution, or its standard deviation, can be calculated using the formula σ = sqrt( (p * (1-p)) / n ), where p is the population proportion and n is the sample size. A sketch of the sampling distribution would show a bell-shaped curve with the mean centered at 20% and the spread becoming narrower as the sample size increases, following the empirical rule.
b. To determine the probability that the sample proportion who die would be .213 or higher, you can use the normal distribution. First, calculate the standard deviation of the sampling distribution using the formula mentioned before: σ = sqrt( (0.20 * (1-0.20)) / 371 ). Then, use a normal probability calculator or a standard normal table to find the probability of getting a sample proportion of .213 or higher. This probability represents the likelihood of observing a sample proportion as extreme as .213 or higher, assuming the population proportion is 20%.
4. If you flipped a coin multiple times and got 75% heads, it would be reasonable to conclude that the coin is not fair. A fair coin has a 50/50 chance of landing heads or tails. To further confirm this, additional information could be asked for, such as the total number of flips or the number of trials conducted. This is necessary because a small number of flips or trials may produce a result that deviates from the expected 50/50 distribution due to random chance. However, if a large number of flips or trials were conducted and consistently resulted in a significantly high percentage of heads, it would provide strong evidence that the coin is not fair.
5. The distribution of house prices being skewed to the right implies that there are a few houses with very high prices while most houses have modest prices. When taking a random sample of 1000 houses, it would not be reasonable to expect the distribution of house prices to be approximately normal. The reason is that a random sample may not fully capture the extreme values or outliers present in the population. The sample distribution may still retain some level of skewness, though it is likely to be less pronounced than the population distribution. However, as the sample size increases, the sample distribution may become more symmetric or bell-shaped, aligning with the properties of the Central Limit Theorem.
6. The expected value of the number of flips required until a fair coin lands heads up for the first time is 2. This means that, on average, it would take two flips to get a heads. It does not mean that exactly two flips will always be needed, but rather that if you were to repeat the experiment many times, the average number of flips needed will converge to 2. In the context of this problem, the expected value provides a measure of the average or typical number of flips required to observe a specific outcome (in this case, a heads) from a series of repeated experiments.