Studdy AI Logo
Math  /  Data & Statistics

QuestionUntited document -... Traditional | Stats | N... Defense / Stats | NBA.. Utah vs LA Stats \& P... Ivica Zubac /// Stats /... Home - Northern Ess... Time Remaining: 52:24 Jona Midterm Exam: Ch 1(1,2)2(1,2,3)3(1,2,3)4(1,2)5(1,2,3)6(1,2)1(1,2) \mathbf{2 ( 1 , 2 , 3 )} 3(1,2,3) 4(1,2) 5(1,2,3) 6(1,2) Question 29 of 30 (1 point) I Question Attempt: 1 of 1 19 20 ≡21\equiv 21 ≡22\equiv 22 23 24 25 26 ≡27\equiv 27 28 29
Coronary bypass surgery: A healthcare research agency reported that 41%41 \% of people who had coronary bypass surgery in 2008 were over the age of 65. Thirteen coronary bypass patients are sampled.
Part 1 of 2 (a) What is the mean number of people over the age of 65 in a sample of 13 coronary bypass patients? Round the answer to two decimal places.
The mean number of people over the age of 65 is □\square .
Part 2 of 2 (b) What is the standard deviation of the number of people over the age of 65 in a sample of 13 coronary bypass patients? Round the answer to four decimal places.
The standard deviation of the number of people over the age of 65 is □\square .

Studdy Solution

STEP 1

What is this asking? If 41% of heart surgery patients are over 65, how many people over 65 do we expect to see in a group of 13 patients, and how much could this number typically vary? Watch out! Don't mix up the formulas for mean and standard deviation of a binomial distribution!

STEP 2

1. Calculate the mean.
2. Calculate the standard deviation.

STEP 3

We're dealing with a **binomial distribution** here!
We have a fixed number of trials (n=13n = \mathbf{13} patients) and the probability of "success" (being over 65) is the same for each patient (p=0.41p = \mathbf{0.41}).
The mean of a binomial distribution is simply n⋅pn \cdot p.
This makes intuitive sense: it's the number of trials times the probability of success on each trial.

STEP 4

Let's **calculate the mean**: μ=n⋅p=13⋅0.41=5.33\mu = n \cdot p = \mathbf{13} \cdot \mathbf{0.41} = \mathbf{5.33} So, on average, we'd expect to see about **5.33** people over 65 in a group of 13 heart surgery patients.

STEP 5

The **standard deviation** of a binomial distribution is given by the formula σ=n⋅p⋅(1−p)\sigma = \sqrt{n \cdot p \cdot (1 - p)}.
This formula tells us how spread out the data is likely to be.
The (1−p)(1-p) term represents the probability of *not* being over 65.

STEP 6

Let's **plug in our values**: σ=n⋅p⋅(1−p)=13⋅0.41⋅(1−0.41)=13⋅0.41⋅0.59\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{\mathbf{13} \cdot \mathbf{0.41} \cdot (1 - \mathbf{0.41})} = \sqrt{\mathbf{13} \cdot \mathbf{0.41} \cdot \mathbf{0.59}}

STEP 7

**Continue the calculation**: σ=3.1357≈1.7708\sigma = \sqrt{\mathbf{3.1357}} \approx \mathbf{1.7708} Rounded to four decimal places, the standard deviation is **1.7708**.
This tells us how much the actual number of patients over 65 is likely to vary around the mean of **5.33**.

STEP 8

(a) The mean number of people over the age of 65 is **5.33**.
(b) The standard deviation of the number of people over the age of 65 is **1.7708**.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord