Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication (Source: I. B. Myers and M. H. McCaulley, A Guide to the Development and Use of the Myers-Briggs Type Indicators). Similarities and Differences in a Random Sample of 375 Married Couples Number of Similar Preferences Number of Married Couples All four Three Two One None 27 126 122 65 35 Suppose that a married couple is selected at random. (a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common. 0 1 2 3 4 (b) Do the probabilities add up to 1? Why should they? Yes, because they do not cover the entire sample space.No, because they do not cover the entire sample space. Yes, because they cover the entire sample space.No, because they cover the entire sample space. (c) What is the sample space in this problem? 0, 1, 2, 3 personality preferences in common 1, 2, 3, 4 personality preferences in common 0, 1, 2, 3, 4, 5 personality preferences in common 0, 1, 2, 3, 4 personality preferences in common

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section: Chapter Questions

Problem 13PT

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Question

1.Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication (Source: I. B. Myers and M. H. McCaulley, A Guide to the Development and Use of the Myers-Briggs Type Indicators).

Similarities and Differences in a Random Sample of 375 Married Couples
Number of Similar Preferences	Number of Married Couples
All four Three Two One None	27 126 122 65 35

Suppose that a married couple is selected at random.

(a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common.

0	1	2	3	4

(b) Do the probabilities add up to 1? Why should they?

Yes, because they do not cover the entire sample space.No, because they do not cover the entire sample space. Yes, because they cover the entire sample space.No, because they cover the entire sample space.

0, 1, 2, 3 personality preferences in common

1, 2, 3, 4 personality preferences in common

0, 1, 2, 3, 4, 5 personality preferences in common

0, 1, 2, 3, 4 personality preferences in common

Expert Solution

Step 1

In this case, the relative frequency formula can be used to find the desired probabilities. The formula to calculate the relative frequency is shown below:

$R e l a t i v e f r e q u e n c y = \frac{f}{n}$

Here, f is the frequency of the occurrence of an event in a sample and n is the sample size.

Substitute the provided values in the above formula to find the required probabilities.

The probability that the married couples will have no personality preferences in common can be calculated as:

$\begin{array}{rcl} P (n o s i m i l a r p r e f e r e n c e s) & = & P (0) \\ = & \frac{N u m b e r o f c o u p l e s w i t h n o s i m i l a r p r e f e r e n c e s}{S a m p l e s i z e} \\ = & \frac{35}{375} \\ = & 0.0933 \end{array}$

The probability that the married couples will have 1 personality preference in common can be calculated as:

$\begin{array}{rcl} P (1 s i m i l a r p r e f e r e n c e s) & = & P (1) \\ = & \frac{N u m b e r o f c o u p l e s w i t h 1 s i m i l a r p r e f e r e n c e s}{S a m p l e s i z e} \\ = & \frac{65}{375} \\ = & 0.1733 \end{array}$

The probability that the married couples will have 2 personality preferences in common can be calculated as:

$\begin{array}{rcl} P (2 s i m i l a r p r e f e r e n c e s) & = & P (2) \\ = & \frac{N u m b e r o f c o u p l e s w i t h 2 s i m i l a r p r e f e r e n c e s}{S a m p l e s i z e} \\ = & \frac{122}{375} \\ = & 0.3253 \end{array}$

The probability that the married couples will have 3 personality preferences in common can be calculated as:

$\begin{array}{rcl} P (3 s i m i l a r p r e f e r e n c e s) & = & P (3) \\ = & \frac{N u m b e r o f c o u p l e s w i t h 3 s i m i l a r p r e f e r e n c e s}{S a m p l e s i z e} \\ = & \frac{126}{375} \\ = & 0.3360 \end{array}$

The probability that the married couples will have 4 personality preferences in common can be calculated as:

$\begin{array}{rcl} P (4 s i m i l a r p r e f e r e n c e s) & = & P (4) \\ = & \frac{N u m b e r o f c o u p l e s w i t h 3 s i m i l a r p r e f e r e n c e s}{S a m p l e s i z e} \\ = & \frac{27}{375} \\ = & 0.0720 \end{array}$

The estimated probability of the married couples having no, one, two, three or four personality matches are given below:

x	0	1	2	3	4
P(x)	0.0933	0.1733	0.3253	0.336	0.072

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