What are the last three terms of the binomial expansion 32x³ + 80x'y + 80x³y² +
1 answer:
The last three terms of the given binomial expansion are 40x²y³ + 10xy⁴ + y⁵
<h3>Binomial expansion</h3>From the question, we are to determine the last three terms of the binomial expansion 32x⁵ + 80x⁴y + 80x³y² + __ + __ + __
The highest power of the expansion is 5
Using the binomial expansion theorem
(m +n)⁵ = m⁵ + 5m⁴n + 10m³n² + 10m²n³ + 5mn⁴ + n⁵
Now, we will determine the values of m and n by comparison
By comparison,
m⁵ = 32x⁵
m⁵ = (2x)⁵
∴ m = 2x
Also,
5m⁴n = 80x⁴y
Divide both sides by 5, to get
m⁴n = 16x⁴y
Put m = 2x into the equation
(2x)⁴n = 16x⁴y
16x⁴n = 16x⁴y
∴ n = y
Thus, the expression being expanded is (2x + y)⁵
By binomial expansion, the expansion of (2x + y)⁵ is
(2x + y)⁵ = (2x)⁵ + 5(2x)⁴y + 10(2x)³(y)² + 10(2x)²(y)³ + 5(2x)(y)⁴ + y⁵
(2x + y)⁵ = 32x⁵ + (5×16)x⁴y + (10×8)x³(y)² + (10×4)x²(y)³ + (5×2)x(y)⁴ + y⁵
(2x + y)⁵ = 32x⁵ + 80x⁴y + 80x³y² + 40x²y³ + 10xy⁴ + y⁵
Hence, the last three terms of the given binomial expansion are 40x²y³ + 10xy⁴ + y⁵
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Answer:
So the p value obtained was a very low value and using the significance level given we have
so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of people that regularly watch the TV station’s news program is significantly less than 0.38 .
Step-by-step explanation:
1) Data given and notation
n=830 represent the random sample taken
X=282 represent the people that regularly watch the TV station’s news program
estimated proportion of people that regularly watch the TV station’s news program
is the value that we want to test
represent the significance level
Confidence=95% or 0.95
z would represent the statistic (variable of interest)
represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the proportion is less than 0.38:
Null hypothesis:
Alternative hypothesis:
When we conduct a proportion test we need to use the z statisitc, and the is given by:
(1)
The One-Sample Proportion Test is used to assess whether a population proportion is significantly different from a hypothesized value
.
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
4) 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 . The next step would be calculate the p value for this test.
Since is a left tailed test the p value would be:
So the p value obtained was a very low value and using the significance level given we have
so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of people that regularly watch the TV station’s news program is significantly less than 0.38 .