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Chapter 6: Problem 33

Is ultrasound a reliable method for determining the gender of an unborn baby? The accompanying data on 1000 births are consistent with summary values that appeared in the online version of the Journal of Statistics Education ("New Approaches to Learning Probability in the First Statistics Course" [2001]). $$ \begin{array}{l|cc} \hline & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{l} \text { Actual Gender Is } \\ \text { Female } \end{array} & 432 & 48 \\ \begin{array}{l} \text { Actual Gender Is } \\ \text { Male } \end{array} & 130 & 390 \\ \hline \end{array} $$ a. Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. b. Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. c. Based on your answers to Parts (a) and (b), do you think ultrasound is equally reliable for predicting gender for boys and for girls? Explain.

Short Answer

Expert verified
The calculation will yield P(Female|Predicted Female) and P(Male|Predicted Male). The comparison of these two probabilities will determine if the ultrasound is equally reliable for predicting both genders. The actual probabilities can only be found by performing the calculations in the above steps.

Step by step solution

01

Calculate the Probability that a Newborn Baby is Female, Given that the Ultrasound Predicted the Baby would be Female

This is a conditional probability: P(Female|Predicted Female). The formula for calculating this is: P(Female ∩ Predicted Female) / P(Predicted Female). From the given table, P(Female ∩ Predicted Female) equals the number of times the actual gender is Female and the ultrasound predicted Female, which is 432. P(Predicted Female) is the total number of times the ultrasound predicted Female, which is the sum of 432 and 130. Therefore, P(Female|Predicted Female) = 432 / (432 + 130).
02

Calculate the Probability that a Newborn Baby is Male, Given that the Ultrasound Predicted the Baby would be Male

Similarly, calculate the conditional probability: P(Male|Predicted Male). Using the formula P(Male ∩ Predicted Male) / P(Predicted Male), P(Male ∩ Predicted Male) is the number of times the actual gender is Male and the ultrasound predicted Male, which is 390. P(Predicted Male) is the total number of times the ultrasound predicted Male, which is the sum of 48 and 390. Hence, P(Male|Predicted Male) = 390 / (48 + 390).
03

Compare the Calculated Probabilities to Determine the Reliability of Ultrasound

To determine if ultrasound is equally reliable for predicting both genders, compare the probabilities calculated in Steps 1 and 2. If the probabilities are approximately equal, then the ultrasound is equally reliable. If not, it means that the ultrasound is more reliable for one gender than the other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ultrasound Gender Prediction
Ultrasound technology has become an invaluable tool in prenatal care, with one of its uses being the ability to predict the gender of an unborn baby. But how reliable is this method? The accuracy of gender prediction via ultrasound depends on several factors, such as the gestational age, the position of the baby, and the skill of the technician.

Considering our exercise dataset, we can apply statistics to assess the reliability of ultrasound for this purpose. Specifically, we'll look at how many times the ultrasound correctly predicted the baby's gender. For instance, if an ultrasound predicted the baby would be female, we determine how many times the prediction was accurate. This is known as conditional probability, and it allows us to express the likelihood of an event occurring given that another event has already occurred. Using the provided data from the 1000 births, we can calculate the probability for both female and male predictions to understand the effectiveness of ultrasounds for gender prediction.
Statistics Education
Statistics education is essential for interpreting data and making informed decisions in various fields. It encompasses the understanding of concepts such as probability, data analysis, and inference. Through statistics, we can extract meaningful insights from data and quantify uncertainty.

This exercise from the Journal of Statistics Education shines a spotlight on probability, a foundational concept in statistics that quantifies how likely an event is to occur. The notion of conditional probability, which is pivotal in this exercise, is particularly useful. It is a measure of the probability of an event given that another event has already occurred. In our scenario, it helps students understand the likelihood that an ultrasound's prediction of a baby's gender is correct, serving as a practical example of how statistics can be applied to real-world situations.
Probability Estimation
Probability estimation is a core aspect of statistics that involves determining the likelihood of a particular event occurring. In the context of our exercise, we computed the conditional probabilities for a baby being female or male given the ultrasound predictions.

To estimate these probabilities, we used the formula for conditional probability, which is the probability of the intersection of two events divided by the probability of the condition event. The calculations in the exercise reveal that the probability of a newborn baby being female, given that the ultrasound predicted female, was \( P(Female \vert Predicted Female) = \frac{432}{432 + 130} \), and the probability of being male, given that the ultrasound predicted male, was \( P(Male \vert Predicted Male) = \frac{390}{48 + 390} \). These estimations let us assess the accuracy of ultrasound predictions and understand the concept of probability estimation better. Through exercises like these, students not only learn the mathematical formulae but also the reasoning and interpretation that are vital in applying these concepts to everyday phenomena.

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Most popular questions from this chapter

Insurance status - covered (C) or not covered (N) \- is determined for each individual arriving for treatment at a hospital's emergency room. Consider the chance experiment in which this determination is made for two randomly selected patients. The simple events are \(O_{1}=(\mathrm{C}, \mathrm{C})\) \(O_{2}=(\mathrm{C}, \mathrm{N}), O_{3}=(\mathrm{N}, \mathrm{C})\), and \(O_{4}=(\mathrm{N}, \mathrm{N}) .\) Suppose that probabilities are \(P\left(O_{1}\right)=.81, P\left(O_{2}\right)=.09, P\left(O_{3}\right)=.09\), and \(P\left(O_{4}\right)=.01\). a. What outcomes are contained in \(A\), the event that at most one patient is covered, and what is \(P(A)\) ? b. What outcomes are contained in \(B\), the event that the two patients have the same status with respect to coverage, and what is \(P(B)\) ?

Consider the chance experiment in which an automobile is selected and both the number of defective headlights \((0,1\), or 2\()\) and the number of defective tires \((0,1\), 2,3, or 4 ) are determined. a. Display possible outcomes using a tree diagram. b. Let \(A\) be the event that at most one headlight is defective and \(B\) be the event that at most one tire is defective. What outcomes are in \(A^{C} ?\) in \(A \cup B ?\) in \(A \cap B ?\) c. Let \(C\) denote the event that all four tires are defective. Are \(A\) and \(C\) disjoint events? Are \(B\) and \(C\) disjoint?

The newspaper article "Folic Acid Might Reduce Risk of Down Syndrome" (USA Today, September 29 , 1999) makes the following statement: "Older women are at a greater risk of giving birth to a baby with Down Syndrome than are younger women. But younger women are more fertile, so most children with Down Syndrome are born to mothers under \(30 .\) " Let \(D=\) event that a randomly selected baby is born with Down Syndrome and \(Y=\) event that a randomly selected baby is born to a young mother (under age 30 ). For each of the following probability statements, indicate whether the statement is consistent with the quote from the article, and if not, explain why not. a. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) b. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.7\) c. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.7\) d. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\) e. \(P(D \mid Y)=.001, \quad P\left(D \mid Y^{C}\right)=.001, \quad P(Y)=.4\) f. \(P(D \mid Y)=.004, \quad P\left(D \mid Y^{C}\right)=.004, \quad P(Y)=.4\)

At a large university, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 students used a book by Professor Mean; during the winter quarter, 300 students used a book by Professor Median; and during the spring quarter, 200 students used a book by Professor Mode. A survey at the end of each quarter showed that 200 students were satisfied with the text in the fall quarter, 150 in the winter quarter, and 160 in the spring quarter. a. If a student who took statistics during one of these three quarters is selected at random, what is the probability that the student was satisfied with the textbook? b. If a randomly selected student reports being satisfied with the book, is the student most likely to have used the book by Mean, Median, or Mode? Who is the least likely author? (Hint: Use Bayes' rule to compute three probabilities.)

Many cities regulate the number of taxi licenses, and there is a great deal of competition for both new and existing licenses. Suppose that a city has decided to sell 10 new licenses for \(\$ 25,000\) each. A lottery will be held to determine who gets the licenses, and no one may request more than three licenses. Twenty individuals and taxi companies have entered the lottery. Six of the 20 entries are requests for 3 licenses, 9 are requests for 2 licenses, and the rest are requests for a single license. The city will select requests at random, filling as many of the requests as possible. For example, the city might fill requests for \(2,3,1\), and 3 licenses and then select a request for \(3 .\) Because there is only one license left, the last request selected would receive a license, but only one. a. An individual has put in a request for a single license. Use simulation to approximate the probability that the request will be granted. Perform at least 20 simulated lotteries (more is better!). b. Do you think that this is a fair way of distributing licenses? Can you propose an alternative procedure for distribution?
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