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astra-53 [7]
3 years ago
8

Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make indepe

ndent choices with P(A) = 0.6, P(B) = 0.3, and P(C) = 0.1.
(a) Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?
Mathematics
1 answer:
Maurinko [17]3 years ago
7 0

Answer:

Mean = 20

Variance = 16

Step-by-step explanation:

Solution:-

- Let X be the number of customers paying with a debit card.  X has the binomial distribution with n = 100 trials and success probability p = 0.2  

- In general, if X has the binomial distribution with (n) trials and a success probability of (p) then:

                 P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

                 for values of x = 0, 1, 2, ..., n  

                 P[X = x] = 0 for any other value of x.

- The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.  

- Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

                  X ~ Binomial( n = 100 , p = 0.2 )

- The mean of the binomial distribution is n * p = 20  

- The variance of the binomial distribution is n * p * (1 - p) = 16  

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Step-by-step explanation:

The amortization formula can be used to find the payment value.

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The value of A in this scenario is the semi-annual payment.

Initially, the interest rate is 0.055 and the number of years remaining is 25.

After the first 5-year period, the interest rate goes up by 12%, so is multiplied by a factor of 1.12 to become 6.16% per year. The same growth factor is applied to the interest rate at the beginning of each 5-year period.

Of course, the number of years remaining is decreased by 5 years at the beginning of the next 5-year period.

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The Principal remaining at the end of each 5-year period is the starting principal for the next period. It is calculated from ...

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Where P is the starting principal, A is the loan payment as calculated above, and r is the annual interest rate.

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These formulas are built into spreadsheet functions, so the desired set of loan payments can be calculated easily by that technology. The result is attached. In the "# pmts" column is the value used to amortize the loan for the 5-year period. The semiannual payment is calculated as though the loan would be completely paid off in that number of payments, keeping the same interest rate for the duration. Of course, that payment series is interrupted and the loan recalculated at the beginning of the next 5 years.

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  $35,176.68

  $36,869.23

  $38,266.75

  $39,158.08

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<em>Additional comment</em>

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Answer:

t=\frac{201-190}{\frac{10}{\sqrt{25}}}=5.5    

p_v =P(t_{(24)}>5.5)=0.00000589  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is higher than 190

Step-by-step explanation:

Data given and notation  

\bar X=201 represent the mean

s=10 represent the sample standard deviation for the sample  

n=25 sample size  

\mu_o =190 represent the value that we want to test

\alpha=0.05 represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

p_v represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 190, the system of hypothesis would be:  

Null hypothesis:\mu \leq 190  

Alternative hypothesis:\mu > 190  

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

t=\frac{201-190}{\frac{10}{\sqrt{25}}}=5.5    

P-value

The first step is calculate the degrees of freedom, on this case:  

df=n-1=25-1=24  

Since is a one side test the p value would be:  

p_v =P(t_{(24)}>5.5)=0.00000589  

Conclusion  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that the true mean is higher than 190

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3 years ago
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