I tried something similar to the notation of (x+2)^7, etc, did not get close at all, how would this be solved?
Answers
[tex] 24 = 3 \cdot 2^3 [/tex]
[tex]96=3\cdot 2^5 [/tex]
[tex] 384=3\cdot2^7[/tex]
hence it is a geometric progression, with a multiplied constant [tex]3[/tex]
Sum of G.P. of [tex]n[/tex] terms [tex] S_n = a\dfrac{r^n-1}{r-1}\quad \text{where } r \text{ is the common ratio and } a \text{ is the first term} [/tex]
and [tex] r=-2^2=-4[/tex]
Note that the constant should be separated, so
[tex] a= -8 [\tex]
after plugging the values, you'll get the answer
[tex] -26216 \times 3 [/tex]
which option C
Answer:
C
Step-by-step explanation:
-24+96-384+...
a=-24
r=96/(-24)=-4
[tex]s_{7}=a\frac{1-r^7}{1-r} \\=-24\frac{1-(-4)^7}{1-(-4)}\\=-24\frac{1+4^7}{1+4} \\=-24\frac{1+16384}{5} \\=-24\frac{16385}{5} \\=-24 \times 3277\\=-78648[/tex]
Related Questions
find the greatest number that divides 56 and 84 exactly
Answers
Answer:
28
Step-by-step explanation:
Find the gcf
Chris wanted to know how likely he is to win at his favorite carnival game. He conducted 50 tests and won 15 times. What is the probability that he will win next time he plays? All answers are rounded to the nearest hundredth. a.) 0.15 b.) 0.30 c.) 0.50 d.) 0.35 SUBMIT MY ANSWER g
Answers
Answer:
b.) 0.30
Step-by-step explanation:
15/50 = 0.3
Simplify (x + 4)(x2 − 6x + 3). x3 − 14x2 + 3x + 12 x3 − 6x2 − 17x + 12 x3 − 10x2 − 27x + 12 x3 − 2x2 − 21x + 12
Answers
Answer:
36 x^3 - 32 x^2 + (x + 4) (x^2 - 6 x + 3).x^3 - 62 x + 12
Step-by-step explanation:
Answer:
x^6-2x^5-21x^4+48x^3-32x^2-62x+12
Step-by-step explanation:
Mark me as brainliest!!!!
Sean earned 20 points. Charles earned p more points than Sean. Choose the expression that shows how many points Charles earned.
Answers
Therefore, the expression to show how many points Charles earned would be:
p + 20
Answer:
the person above is correct if i did this correct
Step-by-step explanation:
A loaf of bread costs $1.40 and the markup is 30% of the selling price. Find the selling price.
Answers
Answer:
The selling price after the markup is $1.82
Step-by-step explanation:
$1.40 * .30 =
Multiply $1.40 times .30 (which is same as 30%)
$1.40 * .30 = $0.42
Add $1.40 and $0.42
= $1.82
Hope this helps.
how do you change a hot air baloon descends 200 feet per minute from a altitude of 1000 feet into a algebraic expression.
Answers
How would x^3+8 be expressed in factored form
Answers
[tex]\it x^3+8=x^3+2^3=(x+2)(x^2-2x+4)\\ \\ \\ a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]
Find the value of the variable(s) in each figure. Explain your reasoning. Thank you in advance
Answers
Answer:
1. x 55
2. y 117
x 51
3.x39
y116
4.x 18
5.x 48
y 14
for the last one I'm not sure. please give 5 start
Plz answer quick will give good rate and thanksss
h(x) = (x - 3)^2 determine which x-value whether it is in the domain of h or not
In domain not in domain
0
3
4
Answers
Answer:
Hey there!
All of the values: 0, 3, and 4 are in the domain.
This is because h(x) = (x - 3)^2 is a parabola, or a quadratic. By definition, the domain, or the possible x values of a parabola are infinite.
Hope this helps :)
a word problem on proportions using a unit rate
Lashonda made $273 for 13 hours of work.
At the same rate, how many hours would she have to work to make $231?
hours
Х
?
Answers
eleven hours - 11 hours
Step 1: 273/13 = 21
So, x equals 21
Step 2: 231/21 = 11
Write x2 − 2x − 3 = 0 in the form (x − a)2 = b, where a and b are integers. (1 point) (x − 4)2 = 3 (x − 3)2 = 2 (x − 2)2 = 1 (x − 1)2 = 4
Answers
Answer:
Answer is (x - 1)² = 4
Step-by-step explanation:
[tex]{ \tt{ {x}^{2} - 2x - 3 = 0}} \\ [/tex]
By completing squares:
[tex]{ \tt{ {x}^{2} - 2x + { (\frac{2}{2} )}^{2} - 3 - {( \frac{2}{2}) }^{2} = 0 }} \\ { \tt{( {x}^{2} - 2x + 1) = 4}} \\ { \tt{(x - 1) {}^{2} = 4 }}[/tex]
[tex]{ \underline{ \sf{ \blue{christ \:† \: alone }}}}[/tex]
A person collected $700 on a loan of $600 they made 5 years ago. If the person charged simple interest, what was the rate of interest? The interest rate is %. (Type an integer or decimal rounded to the nearest hundredth as needed.)
Answers
Answer:
Rate= 3 1/3%
Or Rate= 3.33%
Step-by-step explanation:
Final amount collected= $700
Initial amount given out= $600
Interest made= Final amount - initial amount
Interest made= $700-$600
Interest made= $100
Type of interest rate = simple
Number of years = 5
PRT/100= interest
R=(100*interest)/(PT)
R= (100*100)/(600*5)
R= 10000/3000
R= 10/3
R= 3 1/3%
Or R= 3.33%
show working to this question
Answers
Answer:
Step-by-step explanation:
A = {p, q, r}
Subsets: {p,q,r}, {p,q}, {p,r}. {q,r}, {p}, {q}, {r}, ∅
::::
Slope of RT = (-7/2 - 0)/(-3/2 - 0) = 7/3
Point-slope equation for line of slope 7/3 that passes through (0,0):
y = (7/3)x
The amount of money spent on textbooks per year for students is approximately normal.
a. To estimate the population mean, 19 students are randomly selected the sample mean was $390 and the standard deviation was $120. Find a 95% confidence for the population meam.
b. If the confidence level in part a changed from 95% 1to1999%, would the margin of error for the confidence interval (mark one answer): decrease stay the same increase not enough information to answer
c. If the sample size in part a changed from 19 10 22. would the margin of errot for the confidence interval (mark one answer): decrease in stay the same increase in not enough information to answer
d. To estimate the proportion of students who purchase their textbookslused, 500 students were sampled. 210 of these students purchased used textbooks. Find a 99% confidence interval for the proportion of students who purchase used text books.
Answers
Answer:a
a
[tex]336.04 < \mu < 443.96[/tex]
b
The margin of error will increase
c
The margin of error will decreases
d
The 99% confidence interval is [tex]0.4107 < p < 0.4293[/tex]
Step-by-step explanation:
From the question we are told that
The sample size [tex]n = 19[/tex]
The sample mean is [tex]\= x = \$\ 390[/tex]
The standard deviation is [tex]\sigma = \$ \ 120[/tex]
Given that the confidence level is 95% then the level of significance is mathematically represented as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table
So
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
The margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma}{\sqrt{n} }[/tex]
=> [tex]E = 1.96 * \frac{120}{\sqrt{19} }[/tex]
=> [tex]E = 53.96[/tex]
The 95% confidence interval is
[tex]\= x - E < \mu < \= x + E[/tex]
=> [tex]390 - 53.96 < \mu < 390 - 53.96[/tex]
=> [tex]336.04 < \mu < 443.96[/tex]
When the confidence level increases the [tex]Z_{\frac{\alpha }{2} }[/tex] also increases which increases the margin of error hence the confidence level becomes wider
Generally the sample size mathematically varies with margin of error as follows
[tex]n \ \ \alpha \ \ \frac{1}{E^2 }[/tex]
So if the sample size increases the margin of error decrease
The sample proportion is mathematically represented as
[tex]\r p = \frac{210}{500}[/tex]
[tex]\r p = 0.42[/tex]
Given that the confidence level is 0.99 the level of significance is [tex]\alpha = 0.01[/tex]
The critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table is
[tex]Z_{\frac{\alpha }{2} } = 2.58[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} }* \sqrt{ \frac{\r p (1- \r p )}{n} }[/tex]
=> [tex]E = 0.42 * \sqrt{ \frac{0.42 (1- 0.42 )}{ 500} }[/tex]
=> [tex]E = 0.0093[/tex]
The 99% confidence interval is
[tex]\r p - E < p < \r p + E[/tex]
[tex]0.42 - 0.0093 < p < 0.42 + 0.0093[/tex]
[tex]0.4107 < p < 0.4293[/tex]
A department store offers two promotions. Promotion A says, "Buy one pair of shoes, get the second pair for half the price." Promotion B says, "Buy one pair of shoes, get $10 off the second pair." Jane wants to buy two pairs of shoes that cost $30 each. She can only use one of the promotions, A or B. Jane decides to use the promotion that will save her the most money. How many dollars does Jane save by picking one promotion over the other? (For example, if Jane spends $150 on her purchase by using one promotion and $100 on her purchase by using the other promotion, she saves $150-100=50$ dollars by using the second promotion over the first.)
Answers
Answer:
$5
Step-by-step explanation:
Using Promotion A, Jane would buy the first pair for $30 and the second for 1/2 * 30 = $15 for a total of 30 + 15 = $45. Using Promotion B, she would buy the first pair for $30 and the second for 30 - 10 = $20 for a total of 30 + 20 = $50. Since 45 < 50, Promotion A is the better deal, so Jane would save 50 - 45 = $5.
Carl recorded the number of customers who visited his new store during the week:
Day Customers
Monday 17
Tuesday 13
Wednesday 14
Thursday 16
He expected to have 15 customers each day. To answer whether the number of customers follows a uniform distribution, a chi-square test for goodness of fit should be performed. (alpha = 0.10)
What is the chi-squared test statistic? Answers are rounded to the nearest hundredth.
Answers
Answer:
The chi - square test can be [tex]\approx[/tex] 0.667
Step-by-step explanation:
From the given data :
The null hypothesis and the alternative hypothesis can be computed as:
Null hypothesis: The number of customers does follow a uniform distribution
Alternative hypothesis: The number of customers does not follow a uniform distribution
We learnt that: Carl recorded the number of customers who visited his new store during the week:
Day Customers
Monday 17
Tuesday 13
Wednesday 14
Thursday 16
The above given data was the observed value.
However, the question progress by stating that : He expected to have 15 customers each day.
Now; we can have an expected value for each customer as:
Observed Value Expected Value
Day Customers
Monday 17 15
Tuesday 13 15
Wednesday 14 15
Thursday 16 15
The Chi square corresponding to each data can be determined by using the formula:
[tex]Chi -square = \dfrac{(observed \ value - expected \ value )^2}{expected \ value}[/tex]
For Monday:
[tex]Chi -square = \dfrac{(17 - 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(2)^2}{15}[/tex]
[tex]Chi - square = \dfrac{4}{15}[/tex]
chi - square = 0.2666666667
For Tuesday :
[tex]Chi -square = \dfrac{(13- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(-2)^2}{15}[/tex]
[tex]Chi - square = \dfrac{4}{15}[/tex]
chi - square = 0.2666666667
For Wednesday :
[tex]Chi -square = \dfrac{(14- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(-1 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )}{15}[/tex]
chi - square = 0.06666666667
For Thursday:
[tex]Chi -square = \dfrac{(16- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )}{15}[/tex]
chi - square = 0.06666666667
Observed Value Expected Value chi - square
Day Customers
Monday 17 15 0.2666666667
Tuesday 13 15 0.2666666667
Wednesday 14 15 0.06666666667
Thursday 16 15 0.06666666667
Total : 0.6666666668
The chi - square test can be [tex]\approx[/tex] 0.667
At level of significance ∝ = 0.10
degree of freedom = n - 1
degree of freedom = 4 - 1
degree of freedom = 3
At ∝ = 0.10 and df = 3
The p - value for the chi - square test statistics is 0.880937
Decision rule: If the p - value is greater than the level of significance , we fail to reject the null hypothesis
Conclusion: Since the p - value is greater than the level of significance , we fail to reject the null hypothesis and conclude that there is insufficient evidence to show that the number of customers does not follows a uniform distribution.
Answer:.67
Step-by-step explanation:
If f(x)=logx, show that f(x+h)-f(x)/h=log[1+h/x]^1/h, h=/=0 (Picture attached, thank you!)
Answers
Answer:
Step by step proof shown below.
Step-by-step explanation:
To prove the equation, you need to apply the Logarithm quotient rule and the Logarithm power rule. Here's how the quotient rule looks like.
[tex]log_b(x/y) = log_b(x) - log_b(y)[/tex]
And here's how the power rule looks like
[tex]log_a(x)^n = nlog_a(x)[/tex]
First let's apply the quotient rule.
[tex]\frac{f(x+h)-f(x)}{h} = \frac{log_a(x+h)-log_a(x) }{h} = \frac{log_a(\frac{x+h}{x} )}{h}[/tex]
Now we can do some quick simplification, and apply the power rule.
[tex]\frac{1}{h} log_a(1 + \frac{h}{x} ) = log_a(1+\frac{h}{x} )^\frac{1}{h}[/tex]
An expression is shown below:
f(x) = −16x2 + 22x + 3
Part A: What are the x-intercepts of the graph of the f(x)? Show your work.
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work.
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph.
Answers
Answer:
Part A:
0 = -16x^2 + 22x + 3
x intercepts = (-0.125, 0) and (1.5, 0)
Part B:
It's going to be a maximum. This is because, the graph opens to the bottom. The coordinates are approximately (0.69, 10.56).
Part C:
I would plug in various x values, then solve for the y value. Repeat the process, and get a good idea of the graph.
Hope this helps!
Write
801
1000
as a decimal number.
Answers
Answer:
0.801
Step-by-step explanation:
Answer:
0.801
Step-by-step explanation:
801/1000 = 0.801
HELP ASAP
What is the area of the circle shown below?
Answers
Answer:
C
Step-by-step explanation:
The area (A) of a circle is calculated as
A = πr² ( r is the radius )
Here r = 18 cm , thus
A = π × 18² = 324π ≈ 1017.9 cm² → C
Answer:
C.) 1017.9 cm²
Step-by-step explanation:
For a given circle
radius (r) = 18 cm
Now,
Area of Circle
= πr²
= 3.14 × (18)² cm
= 3.14 × 324 cm
= 1017.9 cm²
Business/multivariable calc question
help needed asap!!!!
Answers
Answer:
There is a min value of 376 located at (x,y) = (9,7)
============================================================
Explanation:
Solve the second equation for y
x+y = 16
y = 16-x
Then plug it into the first equation
f(x,y) = 3x^2+4y^2 - xy
g(x) = 3x^2+4(16-x)^2 - x(16-x)
g(x) = 3x^2+4(256 - 32x + x^2) - 16x + x^2
g(x) = 3x^2+1024 - 128x + 4x^2 - 16x + x^2
g(x) = 8x^2-144x+1024
The positive leading coefficient 8 tells us we have a parabola that opens upward, and produces a minimum value (aka lowest point) at the vertex.
Let's compute the derivative and set it equal to zero to solve for x.
g(x) = 8x^2-144x+1024
g ' (x) = 16x-144
16x-144 = 0
16x = 144
x = 144/16
x = 9
The min value occurs when x = 9. Let's find its paired y value.
y = 16-x
y = 16-9
y = 7
The min value occurs at (x,y) = (9,7)
Lastly, let's find the actual min value of f(x,y).
f(x,y) = 3x^2+4y^2 - xy
f(9,7) = 3(9)^2+4(7)^2 - 9*7
f(9,7) = 376
The smallest f(x,y) value is 376.
· f(x)= x2 - 49
Identify the number of zeros of the polynomial function
Answers
Answer:
x = -7, x = 7
Step-by-step explanation:
Firstly, you are going to set the equation to 0, and then factor it.
Set equation to 0 -----> f(x)= x^2 - 49 will become x^2 - 49 = 0
Now, you're going to factor the equation.
You'll get (x-7) (x+7) upon factoring.
Thirdly, you will set (x-7)(x+7) equal to 0 and also solve for x.
Keep in mind that you'll be treating them as two separate equations
So, ----> (x-7) = 0 (x+7) = 0
When you solve for the x, you'll find out that x is equal to 7 and -7 ---> these are your zeros.
which table shows a proportional relationship between x and y?
Answers
Answer:
Table C
Step-by-step explanation:
For x and y to be proportional , then the values of
[tex]\frac{y}{x}[/tex] = constant k
Table B
[tex]\frac{y}{x}[/tex] = [tex]\frac{6}{3}[/tex] = 2
[tex]\frac{y}{x}[/tex] = [tex]\frac{24}{6}[/tex] = 4
[tex]\frac{y}{x}[/tex] = [tex]\frac{36}{9}[/tex] = 4
The values are not constant
Table C
[tex]\frac{y}{x}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{y}{x}[/tex] = [tex]\frac{4}{6}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{y}{x}[/tex] = [tex]\frac{6}{9}[/tex] = [tex]\frac{2}{3}[/tex]
These values are constant
Then Table C shows a proportional relationship between x and y
Find x in each triangle. PLZ ANSWER FAST!!!!!!!!!!!
Answers
c^2 - a^2 = b^2
13^2 - 5^2 = b^2
b = 12dm
b) a^2 + b^2 = c^2
10^2 + 24^2 = c^2
c = 26 in
c) a^2 + b^2 = c^2
c^2 - a^2 = b^2
17^2 - 8^2 = b^2
b = 15 ft
Define “constant value”
Answers
A fixed value. In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number. Example: in "x + 5 = 9", 5 and 9 are constants.
Use De Moivre's theorem to find the indicated power of the complex number. Write the answer in rectangular form.[2(cos10∘ + i sin10∘)]^3.
Answers
Answer:
[tex]\bold{4\sqrt3 + i4}[/tex]
Step-by-step explanation:
Given complex number is:
[tex][2(cos10^\circ + i sin10^\circ)]^3[/tex]
To find:
Answer in rectangular form after using De Moivre's theorem = ?
i.e. the form [tex]a+ib[/tex] (not in forms of angles)
Solution:
De Moivre's theorem provides us a way of solving the powers of complex numbers written in polar form.
As per De Moivre's theorem:
[tex](cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))[/tex]
So, the given complex number can be written as:
[tex][2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3[/tex]
Now, using De Moivre's theorem:
[tex]\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}[/tex]
So, the answer in rectangular form is:
[tex]\bold{4\sqrt3 + i4}[/tex]
HCF of x minus 2 and X square + X - 6
Answers
Answer:
[tex] \boxed{ \sf{ \bold{ \huge{ \boxed{x - 2}}}}}[/tex]Step-by-step explanation:
[tex] \sf{x - 2} \: and \: { {x}^{2} + x - 6}[/tex]
To find the H.C.F of the algebraic expressions, they are to be factorised and a common factor or the product of common factors is obtained as their H.C.F
Let's solve
First expression = x - 2
Second expression = x + x - 6
Here, we have to find the two numbers which subtracts to 1 and multiplies to 6
= x + ( 3 - 2 ) x + 6
Distribute x through the parentheses
= x + 3x - 2x + 6
Factor out x from the expression
= x ( x + 3 ) - 2x + 6
Factor out -2 from the expression
= x ( x + 3 ) - 2 ( x + 3 )
Factor out x+3 from the expression
= ( x + 3 ) ( x - 2 )
Here, x - 2 is common in both expression.
Thus, H.C.F = x - 2
Hope I helped!
Best regards!!!
Answer:
x - 2
Step-by-step explanation:
by factorization method
1) x - 2
2) x^2 + x - 6
by splitting method
x^2 + 3x - 2x - 6
taking separate common from the first two terms and last two terms
x(x + 3) - 2(x + 3)
now writing x+3 once and the other term to get the right answer
(x + 3)(x - 2)
in both parts just see the similar term and write it as HCF
HCF= x - 2
and the second method by which you can get this answer is division method
Salina currently has an account balance of $1,047.69. Her initial deposit on the account was $630 and it earned 3.9% simple interest. How long has Salina held the account?
A - 17 years
B - 26 years
C - 10 years
D - 43 years
Answers
Answer:
A. 17 years
Step-by-step explanation:
Use the simple interest equation, I = prt, where I is the interest money gained, p is the starting amount of money, r is the interest rate in decimal form, and t is the time in years.
Plug in the values to solve for t:
417.69 = (630)(0.039)(t)
417.69 = 24.57t
17 = t
= 17 years
So, the correct answer is A, 17 years
The following sample contains the scores of 6 students selected at random in Mathematics and English. Use the scores in English as the dependent variable Y.
Mathematics score (X) 70 92 80 74 65 83
English score 74 84 63 87 78 90
Σx =464 Σy=476 Σx^2= 36354 Σy^2=38254 Σxy= 36926
Find the sample coefficient of determination and interpret.
a. 0.0575 and prediction accuracy is 5.75%
b. 0.2397 and prediction accuracy is 23.97%
c. 0.0575 and prediction accuracy is 94.25%
d. 0.2397 and prediction accuracy is 76.03%
Answers
Answer:
d the answer is d
Step-by-step explanation:
Beer shelf life is a problem for brewers and distributors because when beer is stored at room temperature, its flavor deteriorates. When the average furfuryl ether content reaches 6 μg per liter, a typical consumer begins to taste an unpleasant chemical flavor. At α = .05, would the following sample of 12 randomly chosen bottles stored for a month convince you that the mean furfuryl ether content exceeds the taste threshhold? 8.92 6.99 5.54 5.73 6.38 5.51 6.45 7.50 8.48 5.56 6.90 6.46
Answers
Answer:
As the calculated value of t =2.1698 is greater than t (0.05,11) = 1.796 reject H0 . It means chosen bottles stored for a month convince you that the mean furfuryl ether content exceeds the taste threshhold.
Step-by-step explanation:
We formulate our null and alternative hypotheses as
H0 u≤ 6 ug Ha : u > 6 ug
The significance level ∝ = 0.05
The test statistic used is
t = X` - u / s/ √n
which if H0 is true, has the students' t test with n-1 = 11 degrees of freedom.
The critical region t > t (0.05,11) = 1.796
We compute the t value from the data
Xi Xi²
8.92 79.5664
6.99 48.8601
5.54 30.6916
5.73 32.8329
6.38 40.7044
5.51 30.3601
6.45 41.6025
7.50 56.25
8.48 71.9104
5.56 30.9136
6.90 47.61
6.46 41.7316
80.42 553.0336
Now x` = ∑x/ n = 80.42/12 = 6.70
S²= 1/n-1 ( ∑(xi- x`)²= 1/11 ( 553.034 - (80.42)²/12)
= 1/11 (553.034-538.948) = 1.2805
s= 1.1316
Putting the values in the test statistics
t = X` - u / s/ √n = 6.70- 6 / 1.1316 / √12
= 2.1698
The critical region t > t (0.05,11) = 1.796
As the calculated value of t =2.1698 is greater than t (0.05,11) = 1.796 reject H0 . It means chosen bottles stored for a month convince you that the mean furfuryl ether content exceeds the taste threshhold.