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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 鈭� Predicted Female) / P(Predicted Female). From the given table, P(Female 鈭� 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 鈭� Predicted Male) / P(Predicted Male), P(Male 鈭� 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

An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose that the same cola has actually been put into all three glasses. a. What are the simple events in this chance experiment, and what probability would you assign to each one? b. What is the probability that \(\mathrm{C}\) is ranked first? c. What is the probability that \(\mathrm{C}\) is ranked first \(a n d \mathrm{D}\) is ranked last?

Suppose that, starting at a certain time, batteries coming off an assembly line are examined one by one to see whether they are defective (let \(\mathrm{D}=\) defective and \(\mathrm{N}=\) not defective). The chance experiment terminates as soon as a nondefective battery is obtained. a. Give five possible experimental outcomes. b. What can be said about the number of outcomes in the sample space? c. What outcomes are in the event \(E\), that the number of batteries examined is an even number?

The student council for a school of science and math has one representative from each of the five academic departments: biology (B), chemistry (C), mathematics (M), physics (P), and statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee (by placing five slips of paper in a bowl, mixing, and drawing out two of them). a. What are the 10 possible outcomes (simple events)? b. From the description of the selection process, all outcomes are equally likely; what is the probability of each simple event? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?

A radio station that plays classical music has a "by request" program each Saturday evening. The percentages of requests for composers on a particular night are as follows: \(\begin{array}{lr}\text { Bach } & 5 \% \\ \text { Beethoven } & 26 \% \\\ \text { Brahms } & 9 \% \\ \text { Dvorak } & 2 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Mozart } & 21 \% \\ \text { Schubert } & 12 \% \\ \text { Schumann } & 7 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Wagner } & 1 \%\end{array}\) Suppose that one of these requests is to be selected at random. a. What is the probability that the request is for one of the three \(\mathrm{B}^{\prime}\) s? b. What is the probability that the request is not for one of the two S's? c. Neither Bach nor Wagner wrote any symphonies. What is the probability that the request is for a composer who wrote at least one symphony?

A family consisting of three people- \(\mathrm{P}_{1}, \mathrm{P}_{2}\), and \(\mathrm{P}_{3}\) \- belongs to a medical clinic that always has a physician at each of stations 1,2, and 3 . During a certain week, each member of the family visits the clinic exactly once and is randomly assigned to a station. One experimental outcome is \((1,2,1)\), which means that \(\mathrm{P}_{1}\) is assigned to station \(1 .\) \(\mathrm{P}_{2}\) to station 2, and \(\mathrm{P}_{3}\) to station 1 a. List the 27 possible outcomes. (Hint: First list the nine outcomes in which \(\mathrm{P}_{1}\) goes to station 1 , then the nine in which \(\mathrm{P}_{1}\) goes to station 2 , and finally the nine in which \(\mathrm{P}_{1}\) goes to station 3 ; a tree diagram might help.) b. List all outcomes in the event \(A\), that all three people go to the same station. c. List all outcomes in the event \(B\), that all three people go to different stations. d. List all outcomes in the event \(C\), that no one goes to station \(2 .\) e. Identify outcomes in each of the following events: \(B^{C}\), \(C^{C}, A \cup B, A \cap B, A \cap C\)
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