The Centers for Disease Control and Prevention (CDC) is requir...

Question

The Centers for Disease Control and Prevention (CDC) is required by law to publish a report on assisted reproductive technology (ART). ART includes all fertility treatments in which both the egg and the sperm are used. These procedures generally involve removing eggs from a patient’s ovaries, combining them with sperm in the laboratory, and returning them to the patient’s body or giving them to another patient.

Pie chart showing ART cycle results for patients under 35: 52% live births, 42.4% eggs retrieved but no birth, 5.6% eggs not retrieved.

You are helping to prepare a CDC report on young ART patients and select at random 6 ART cycles of patients under 35 years of age for a special review. None of the cycles resulted in a live birth. Your manager feels it is impossible to select at random 10 ART cycles that do not result in a live birth. Use the pie chart at the right and your knowledge of statistics to determine whether your manager is correct.

Bar chart showing live birth rates for ART cycles by age group, with percentages decreasing from 52% for patients 34 and under to 3.2% for 43 and older.

b. What probability distribution do you think best describes the situation? Do you think the distribution of the number of live births is discrete or continuous? Explain your reasoning.

Accepted Answer

All right, hello, everyone. So this question says a national health agency maintains records of various fertility treatments. One common treatment is intrauterine insemination, IUI, which is less invasive than IVF. The agency publishes a report showing the outcomes of IUI cycles based on patient age groups. The chart below shows data for patients under the age of 30. Where the rate of successful pregnancy is 25% and no pregnancy is 75%. You are analyzing the success rate of IUI procedures for a research study. You randomly select 8 IUI cycles from the under 30 age group for a detailed review. None of the selected cycles resulted in a successful pregnancy. Part one of the question states your supervisor argues that it is extremely unlikely to select 8 cycles with 0 successful pregnancies randomly. Determine whether your supervisor's concern is valid. For part two, what type of probability distribution would best model the number of successful pregnancies in such samples? Do you think the distribution of successful outcomes is discrete or continuous and explain your answer. So first, to address the validity of the supervisor's concern, we model the situation based on the binomial distribution, because first of all, each IUI cycle is an independent trial. That is, the outcome of one does not affect the other. And each trial has two outcomes. Success for a successful pregnancy or failure for no pregnancy. And the probability of success. It is constant. It's maintained at 25%. So, we know that 8 cycles were considered, which means that we know that N is equal to 8, and P, as mentioned before, is equal to 0.25, representing the probability of success. Now, we know that X, which is the number of successes in the sample, is equal to 0. So we calculate the probability that all 8 cycles result in no pregnancy. So P Of X equals 0 Is equal to. 8 over 0. Multiplied by P to the power of 0. And multiplied by one subtracted by P. To the power of 8 subtracted by 0. So ultimately this expression simplifies to one subtracted by 0.75. To the power of 8, which equals 0.1001. So based on this, the probability of selecting 8 cycles with no successful pregnancies is about 10% or 10.01%. Now, it's noteworthy that this is not extremely rare, because the common cutoff, if you recall, for unusual events is 5% or 0.05. So because our calculated probability is greater than. 0.05. The supervisor's concern is not strongly justified. The outcome is unlikely, but not extremely unlikely. So now on to the type of distribution and the nature of the data. As mentioned before, the number of successful pregnancies follows a binomial distribution because each trial is independent. There are only two possible outcomes, and there's a fixed number of trials. So As for whether or not the distribution is discrete or continuous. Let's recall the difference. Right, a discrete distribution refers to numbers that are countable. That is, they can only be, or they can only take whole number of values. A continuous distribution on the other hand. Oftentimes refers to values that are measured. So, a continuous distribution or a continuous data set can take on any value, including decimals. Now, because we're counting the number of successful pregnancies in this case, that can only take on a whole number of value, ranging from 0 to 8 specifically. So therefore, our data set is considered discrete. So, as mentioned before, the binomial distribution is ultimately the most appropriate model for the situation. So now, let's summarize the final answer. For part one of the question, the answer is no, because 0.10 is not low enough to be considered unusual. For part two, the most appropriate distribution is binomial. The type is discrete, and the reason is a fixed number of independent trials, constant probability of success, and countable outcomes. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Probability Distribution

Discrete vs. Continuous Variables

Random Sampling

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