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Viktor [21]
3 years ago
9

In a study researchers reported the mean BMI for men 60 and older to be 24.7 with a standard deviation of 3.3 and the mean BMI f

or women 60 and older to be 23.1 with a standard deviation of 3.7. Using these values as the populations mean and standard deviation for men and women respectively, find the probability that the difference in the mean BMI (men-women) for 45 women and 50 men selected independently and at random will exceed 2.1.
Mathematics
1 answer:
erica [24]3 years ago
6 0

Answer:

Probability that the difference in the mean BMI (men-women) for 45 women and 50 men selected independently and at random will exceed 2.1 = 0.2451

Step-by-step explanation:

The central limit theorem helps us to obtain the mean and the standard deviation of any sampling distribution.

Given that the sample was obtained from a normal distribution or an approximately normal distribution & it was obtained using random sampling techniques with each variable independent of one another and with each sample with adequate sample size,

Mean of sampling distribution (μₓ) = Population mean (μ)

Standard deviation of the sampling distribution = σₓ = (σ/√N)

where σ = population mean

N = Sample size

For the 45 women

μₓ = μ = 23.1

σₓ = (σ/√N) = (3.7/√45) = 0.552

For the 50 men

μₓ = μ = 24.7

σₓ = (σ/√N) = (3.3/√50) = 0.467

To find the probability that the difference in the mean BMI (men-women) for 45 women and 50 men selected independently and at random will exceed 2.1, we need to combine the distributions.

New distribution = (BMI of men) - (BMI of women) = x = X₁ - X₂

When independent distributions are combined, the combined mean and combined variance are given through the relation

Combined mean = Σ λᵢμᵢ

(summing all of the distributions in the manner that they are combined)

Combined variance = Σ λᵢ²σᵢ²

(summing all of the distributions in the manner that they are combined)

λ₁ = 1, λ₂ = -1

μ₁ = 24.7, μ₂ = 23.1

σ₁ = 0.467, σ₂ = 0.552

Combined mean = (Mean of men) - (Mean of women) = 24.7 - 23.1 = 1.6

Combined Variance = (1²×0.467²) + [(-1)²×(0.552²)] = 0.522793

Combined standard deviation = √0.522793 = 0.723

Probability that the difference in the mean BMI (men-women) for 45 women and 50 men selected independently and at random will exceed 2.1 = P(x > 2.1)

Note that the resulting distribution from the combination of distributions is still a normal distribution since the distributions combined were normal distributions too.

Hence, we first normalize or standardize 2.1

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ = (2.1 - 1.6)/0.723 = 0.69

To determine the required probability

P(x > 2.1) = P(z > 0.69)

We'll use data from the normal distribution table for these probabilities

P(x > 2.1) = P(z > 0.69) = 1 - P(z ≤ 0.69)

= 1 - 0.7549

= 0.2451

Hope this Helps!!!

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Ashley deposits $10 000 at the end of each month for 6 times at rate of 5% compounded monthly. How much will be in her account
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The future values of Ashley's periodic deposits under the five scenarios are stated in the deposit table as follows:

<h3>Deposit Table:</h3>

Item     Initial       Periodic      Deposit    Interest    Future Value    Interest

         Deposit     Deposit        Period        Rate                                    ($)

a.      $10,000     $10,000     6 months        5%        $61,774.51      $1,774.51

b.     $10,000     $10,000     3 months        5%      $30,502.78       $502.78

c.      $10,000    $10,000     6 months        5%        $51,774.51     $1,522.42

d.    $20,000    $10,000      6 months       5%        $72,201.71     $2,201.71

e.   $10,000     $10,000     6 months       5%/6%  $60,803.78       $803.78

<h3>What is the future value?</h3>

The future value of a periodic investment represents the present value of cash flows compounded at an interest rate into the future.

The future value is computed using the FV formula or factor.

It can also be computed using an online finance calculator as follows:

<h3>Data and Calculations:</h3>

Periodic Deposit = $10,000

Investment period = 6 months

Interest rate = 5% compounded monthly

<h3>a) for illustration:</h3>

N (# of periods) = 6 months

I/Y (Interest per year) = 5%

PV (Present Value) = $0

PMT (Periodic Payment) = $10,000

<u>Results</u>:

FV = $61,774.51

Sum of all periodic payments = $60,000 ($10,000 x 6)

Total Interest = $1,774.51

<h3>Other Special Cases:</h3>

c) Interest on $10,000 for 3 months is $1,522.42 ($1,774.51 - $252.09).

d)                             Future Value      Interest

5% with $20,000   $82,285.04    $2,285.04

5% with  $10,000     (10,083.33)          (83.33)

Total for 6 months   $72,201.71     $2,201.71

e)                            Future Value       Interest

5% for 3 months    $30,502.78       $502.78

6% for 3 months     $30,301.00       $301.00

Total for 6 months $60,803.78      $803.78

Learn more about the computation of future values at brainly.com/question/24703884 and brainly.com/question/12979998

#SPJ1

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