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Helga [31]
2 years ago
12

The number of members over time in an online music sharing club can be modeled by an exponential function. the club started with

2100 members. After 1 month, the club had 2142 members.
a. Write an equation for the number of members as a function of time in months since the club started.
b. What is the parameter b in the equation, and what does it represent in this situation?
c. Will the club have more than 5000 members during its first year? Justify your reasoning.
Mathematics
1 answer:
Paraphin [41]2 years ago
7 0

Answer:

a.) N(x) =  N_{0} e^{bx}

b.) therefore b = \ln \frac{2142}{2100}  =  \ln (1.02) = 0.0198

    b is the rate of increase of number of members

c.) The club will not be able to get more than 5000 members during its first year.

Step-by-step explanation:

i) the club started with 2100 members.

  so we can write N_{0} = 2100.

a.) so we can write the equation as an exponential function given by

  N(x) =  N_{0} e^{bx} where x is in months and b is a constant and N(x) is the number of members in the online music sharing club .

 therefore 2142 = 2100 \times (e^{b\times 1}) = 2100e^{b}

b.) therefore b = \ln \frac{2142}{2100}  =  \ln (1.02) = 0.0198

    b is the rate of increase of number of members

c.) Will the club have more than 5000 members during its first year? Justify your reasoning.

      5000 = 2100e^{0.0198x}

 therefore x  = \frac{1}{0.0198} \ln{(\frac{5000}{2100} )}  =  43.81 \hspace{0.1cm}months

The club will not be able to get more than 5000 members during its first year.

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Answer:

a) We would expect to see 500*0.88=440

b) z=\frac{0.9 -0.88}{\sqrt{\frac{0.88(1-0.88)}{500}}}=1.376  

p_v =2*P(Z>1.376)=0.167  

So the p value obtained was a very high value and using the significance level assumed \alpha=0.05 we have p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the true proportion is not significant different from 0.9.

The p value is a criterion to decide if we reject or not the null hypothesis, when p_v we reject the null hypothesis in other case we FAIL to reject the null hypothesis. And represent the "probability of obtaining the observed results of a test, assuming that the null hypothesis is correct".  

Step-by-step explanation:

Data given and notation

n=500 represent the random sample taken

X=450 represent the people that have the seat belt fastened

\hat p=\frac{450}{500}=0.9 estimated proportion of people that have the seat belt fastened

p_o=0.88 is the value that we want to test

\alpha represent the significance level

z would represent the statistic (variable of interest)

p_v{/tex} represent the p value (variable of interest)  Part aWe would expect to see 500*0.88=440Part bConcepts and formulas to use  We need to conduct a hypothesis in order to test the claim that the true proportion changes fro m 0.88.:  Null hypothesis:[tex]p=0.88  

Alternative hypothesis:p \neq 0.88  

When we conduct a proportion test we need to use the z statisitc, and the is given by:  

z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}} (1)  

The One-Sample Proportion Test is used to assess whether a population proportion \hat p is significantly different from a hypothesized value p_o.

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

z=\frac{0.9 -0.88}{\sqrt{\frac{0.88(1-0.88)}{500}}}=1.376  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The significance level assumed is \alpha=0.05. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

p_v =2*P(Z>1.376)=0.167  

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