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Ymorist [56]
2 years ago
10

Consider the function below. g(x) = 4x − 1 Find the difference quotient below (where h ≠ 0) and simplify your answer. g(x + h) −

g(x) -------------------- h
Mathematics
1 answer:
eimsori [14]2 years ago
3 0

Answer:

The difference quotient is 4.

Step-by-step explanation:

Given that:

g(x) = 4x - 1

To find:

Difference quotient = ?

where h \neq 0

Solution:

Formula for Difference quotient is given as:

\dfrac{g(x+h)-g(x)}{h}

First of all, let us find out g(x+h)

Replacing x with x+h

g(x+h) = 4(x+h)-1 \\\Rightarrow g(x+h) = 4x+4h-1

Now,

g(x+h)-g(x) = (4x+4h-1 )-(4x-1)\\\Rightarrow g(x+h)-g(x) = 4x+4h-1 -4x+1\\\Rightarrow g(x+h)-g(x) = 4h

Putting the above value in:

\dfrac{g(x+h)-g(x)}{h} = \dfrac{4h}{h}

We are given that, h \neq 0

\therefore \dfrac{4h}{h}  = 4

So, the difference quotient is 4.

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Answer:

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Step-by-step explanation:

Data given and notation

n=1655 represent the random sample taken

\hat p=0.021 estimated proportion of interest

p_o=0.012 is the value that we want to test

\alpha=0.01 represent the significance level

Confidence=99% or 0.99

z would represent the statistic (variable of interest)

p_v represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that true proportions is higher than 0.012.:  

Null hypothesis:p \leq 0.012  

Alternative hypothesis:p > 0.012  

When we conduct a proportion test we need to use the z statisitic, 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{0.021 -0.012}{\sqrt{\frac{0.012(1-0.012)}{1655}}}=3.363  

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 provided \alpha=0.01. The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

p_v =P(z>3.363)=0.00039  

So the p value obtained was a very low value and using the significance level given \alpha=0.01 we have p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 1% of significance the proportion of interest is significantly higher than 0.012 (1.2%)

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