Question
A medical researcher surveyed a large group of men and women about whether they take medicine as prescribed. The responses were categorized as never, sometimes, or always. The relative frequency of each category is shown in the table.
(a) One person from those surveyed will be selected at random.
(i) What is the probability that the person selected will be someone whose response is never and who is a woman?
(ii) What is the probability that the person selected will be someone whose response is never or who is a woman?
(iii) What is the probability that the person selected will be someone whose response is never given that the person is a woman?
(b) For the people surveyed, are the events of being a person whose response is never and being a woman independent? Justify your answer.
(c) Assume that, in a large population, the probability that a person will always take medicine as prescribed is 0.54 . If 5 people are selected at random from the population, what is the probability that at least 4 of the people selected will always take medicine as prescribed? Support your answer.
▶️Answer/Explanation
Ans:
a(i) \(P(\) never \(n\) woman \()=0.0636\)
The probability that some ane who’s neper is never and is a werner ir 0.0636
a(ii)$
\begin{aligned}
& P \text { (never woman) }-P(\text { never })+P(\text { woman })-P(\text { never } n \text { woman })=0.12+0.53-0.013 \\
& =0.5864 \text { The probability that you Pl\&c someone who said never } \\
& \text { or is a woman is } 0 . \$ 864 \\
&
\end{aligned}
$
a(iii) $
P(\text { never lawman })=\frac{P(\text { nevernowan })}{P \text { (woman) }}=\frac{0.0636}{0.53}=0.12
$
The probability that you pick sombre who said newer given that they were a woman on is 0.12 .
(b) Yes they are independent since the probability of someone saying never given that they are a women and the probability of som one saying never saying never should be the same in order for the two events to be indipadent. In this cases both probabilities equal 0,12 , hence the events of saying never and being \(a\). women are in de pendent.
(c)\(P(\) at least 4 people take their medicine \()=1-P(\) up to 3 people take their medicine \(=1-\) Binempdf \((5,0.54,3)=\{-0.759=0.241\)
The probability of at least 4 people taking their medicine as prescribed is 0.241,
Question
The manager of a grocery store selected a random sample of 11 customers to investigate the relationship between the number of customers in a checkout line and the time to finish checkout. As soon as the selected customer entered the end of a checkout line, data were collected on the number of customers in line who were in front of the selected customer and the time, in seconds, until the selected customer was finished with the checkout. The data are shown in the following scatterplot along with the corresponding least-squares regression line and computer output.
(a) Identify and interpret in context the estimate of the intercept for the least-squares regression line.
(b) Identify and interpret in context the coefficient of determination, \(r^2\).
(c) One of the data points was determined to be an outlier. Circle the point on the scatterplot and explain why the point is considered an outlier.
▶️Answer/Explanation
Ans:
(a)The estimate of the intercept, which is 72.95 seconds, means that if there are no customers in the line, the predicted time to first checkout is 72.95 secmacs.
(b) The \(c^2\) value of \(73.33 \%\) means that about \(73.33 \%\) of the variation of time to finish checkout, \(y\), can be explained of the least-squares regression line of containers in line, \(x\), and time to fish checkout,\(y\).
This points is considered on outlier because its whee is very far from the predicted value of the least squares regression line. The point’s while is about 100 while, when there are 3 costumers in line, the LSRL predicts a value of about 600 .