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3 years ago

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The population (in millions) of a certain island can be approximated by the function p(x)=50(1.05)^x, where x is the number of y

ears since 2000. in which year will the population reach 200 million? hint: an answer such as 2002.4 would represent 2002. a)2028
b)2015
c)2002
d)2066
please help

2 answers:

stich3 [128]3 years ago

5 0

Answer: a)2028

Step-by-step explanation:

Given: The population (in millions) of a certain island can be approximated by the function $p(x)=50(1.05)^x$ , where x is the number of years since 2000.

To find the year in in which year will the population reach 200 million, substitute p(x)=200, we get

$200=50(1.05)^x\\\\\Rightarrow(1.05)^x=4$

Taking natural log on both the sides , we get

$\ln((1.05)^x=\ln(4)\\\\\Rightarrow\ x(\ln(1.05))=\ln4\\\\\Rightarrow\ x=\dfrac{\ln4}{\ln(1.05})=\dfrac{1.38629}{0.04879}\\\\\Rightarrow\ x=28.4134043861\approx28\text{ years}$

The year will be $2000+28=2028$

Hence, in 2028 the population will reach 200 million.

Yuliya22 [10]3 years ago

3 0

The answer is A)2028 - just took the test

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Null hypothesis: $p_{1} = p_{2}$

Alternative hypothesis: $p_{1} \neq p_{2}$

$z=\frac{0.410-0.350}{\sqrt{0.382(1-0.382)(\frac{1}{1000}+\frac{1}{900})}}=2.688$

$p_v =2*P(Z>2.688)= 0.0072$

Comparing the p value with the significance level assumed $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different at 5% of significance.

Step-by-step explanation:

Data given and notation

$X_{1}=410$ represent the number of people indicating that their financial security was more than fair in 2012

$X_{2}=315$ represent the number of people indicating that their financial security was more than fair in 2010

$n_{1}=1000$ sample 1 selected

$n_{2}=900$ sample 2 selected

$p_{1}=\frac{410}{1000}=0.410$ represent the proportion estimated of people indicating that their financial security was more than fair in 2012

$p_{2}=\frac{315}{900}=0.350$ represent the proportion estimated of people indicating that their financial security was more than fair in 2010

$\hat p$ represent the pooled estimate of p

z would represent the statistic (variable of interest)

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

$\alpha=0.05$ significance level given

Part a: Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference between the two proportions, the system of hypothesis would be:

Null hypothesis: $p_{1} = p_{2}$

Alternative hypothesis: $p_{1} \neq p_{2}$

Hypothesis testing

We need to apply a z test to compare proportions, and the statistic is given by:

$z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$ (1)

Where $\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{410+315}{1000+900}=0.382$

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

Calculate the statistic

Replacing in formula (1) the values obtained we got this:

$z=\frac{0.410-0.350}{\sqrt{0.382(1-0.382)(\frac{1}{1000}+\frac{1}{900})}}=2.688$

Statistical decision

Since is a two sided test the p value would be:

$p_v =2*P(Z>2.688)= 0.0072$

Comparing the p value with the significance level assumed $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different at 5% of significance.

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