The probability that the sample of 50 rabbits contains exactly 2 marked rabbits is 1/2.
Explain about the hypergeometric distribution?When sampling from either a small population without replacement, the hypergeometric distribution consists of a discrete probability distribution which determines the likelihood that an event occurs k times in n trials.
The absence of replacements in the hypergeometric distribution sets it apart from the binomial distribution.As a result, it is frequently used in random sampling to ensure statistical quality. A straightforward, indication would be choosing team members at random from a population of males and girls.The given data:
Total market sample of rabbits = 100.
Randomly selected rabbits = 50.
Let the probability that the sample of 50 rabbits contains exactly 2 marked rabbits be P(E).
Then, using hypergeometric distribution;
P(E) = ⁵⁰C₂ / ¹⁰⁰C₂
Solve the probability using the combination.
P(E) = 25*49 / 50*49
P(E) = 25/50
P(E) = 1/2
Thus, the probability that the sample of 50 rabbits contains exactly 2 marked rabbits is 1/2.
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Need help with this project
The function that represents the graph below is given as follows:
c. y = 4x³.
How to obtain the function which defines the graph?Before defining the function which defines the graph, we must obtain the domain and the range of the function, as follows:
Domain -> set of input values assumed by the function -> x values -> all real values in the graph.Range -> set of output values assumed by the function -> y values -> all real values in the graph.As the function has a range of all real values, we know that it is a cube function and not a square function, removing options a and b.
The function is positive and negative as follows:
Positive: positive values of x.Negative: negative values of x.Hence the function has a positive leading coefficient, meaning that option c is the correct option.
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Find the values of the variables and the measures of the indicated angles.
Answer:
x = 75°
Step-by-step explanation:
In a cyclic quadrilateral, opposite angles are supplementary.
x + 105 = 180
Subtract 105 from both sides,
x = 180 - 105
[tex]\boxed{\bf x = 75^\circ}[/tex]
The no. 1 basketball team made 72% of its free throws over the past 5 seasons and shot 450 free throws in total. The
no. 2 team made 64% of its free throws over the past 5 seasons and shot 550 free throws in total. You randomly select
40 free throws from each team and record whether they made the shot. Let o, -, be the difference in the sample
proportions of successful free throws where o, is the proportion of free throw shots made by the no. 1 team and is
the proportion of free throw shots made by the no. 2 team.
Determine the mean of the sampling distribution of Ô, - Ôz. Enter your answer as a decimal rounded to the
hundredths place.
The mean difference of the sampling distribution of the difference in the proportion of free throw shots made between
the no. 1 and no. 2 teams is
The mean of the sampling distribution of Ô1 - Ô2 is 0.08, which indicates that on average, the no. 1 team makes 8% more free throws than the no. 2 team.
The mean of the sampling distribution of the difference in sample proportions of successful free throws, Ô1 - Ô2, can be calculated as:
mean(Ô1 - Ô2) = Ô1 - Ô2
Since the samples are random and independent, the mean of the sampling distribution is simply the difference between the population proportions:
mean(Ô1 - Ô2) = Ô1 - Ô2 = 0.72 - 0.64 = 0.08
Therefore, the mean of the sampling distribution of Ô1 - Ô2 is 0.08, which indicates that on average, the no. 1 team makes 8% more free throws than the no. 2 team.
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The goal of the exercise is to find the year in which cigarette consumption reach
The exponential function of the scenario is [tex]C(t) = C_0 * e^{(r\times t)}[/tex]
To achieve this goal, we need a mathematical model that relates cigarette consumption to time. One such model is the following exponential function:
[tex]C(t) = C_0 * e^{(r\times t)}[/tex]
Here, C_0 is the initial cigarette consumption, r is the growth rate, and t is the time variable. This model assumes that cigarette consumption grows exponentially over time.
We can use this model to predict cigarette consumption at any given time t.
To find the year in which cigarette consumption reaches C_goal, we need to solve for t_goal in the equation above. That is, we need to find the value of t for which C(t) = C_goal. To do this, we can rearrange the equation above and solve for t:
[tex]t_{goal} = (1/r) \times ln(C_{goal}/C_0)[/tex]
Here, ln denotes the natural logarithm. This equation gives us the value of t for which cigarette consumption reaches the goal amount C_goal.
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Complete Question:
The goal of the exercise is to find the year in which cigarette consumption reach. Define the function for the given scenario?
It is 7 feet long, 3 1/4 feet deep, And has a volume of 159 1/4 what is the width
The width of the rectangular prism is 7 feet, given the length is 7 feet, the depth is 3 1/4 feet, and the volume is 159 1/4 cubic feet.
We can use the volume of a rectangular prism formula, which is length x width x height. We are given the length (7 feet), the depth (3 1/4 feet), and the volume (159 1/4 cubic feet). Let's substitute these values into the formula and solve for the width:
7 feet x width x 3 1/4 feet = 159 1/4 cubic feet
Multiplying 7 feet and 3 1/4 feet, we get:
22 3/4 square feet x width = 159 1/4 cubic feet
Dividing both sides by 22 3/4 square feet, we get:
width = (159 1/4 cubic feet) / (22 3/4 square feet)
Simplifying the fraction, we get:
width = (637/4) / (91/4)
As multiplying by a fraction's reciprocal is equivalent to dividing by it, we get:
width = (637/4) x (4/91)
The factor of 4 in the numerator and denominator cancels out, so we get:
width = 637/91
Simplifying the fraction, we get:
width = 7
Therefore, the width of the rectangular prism is 7 feet.
Given the length, depth, and volume of a rectangular prism, we can use the formula for the volume to solve for the width. In this case, we substitute the given values and solve for the unknown width by simplifying the algebraic expression. The resulting width is 7 feet.
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describe the following solid figures
Cube
Rectangular Prism
Pyramid
Cone
Cylinder
Sphere
Answer:
1. Cube - In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
2. Rectangular prism - is a solid figure that has six sides, called faces, that are rectangles.
3. Pyramid - is a solid figure that has a polygon as its base on one end and triangular faces all meeting at a single point on the other end.
4. Cone - a solid figure that has a circular face on one end, called the base, and a point at the other end where the sides meet.
5. Cylinder - is a solid figure that has two circular bases and one curved side.
6. Sphere - is a solid figure that is round and has the shape of a ball.
Step-by-step explanation:
In a hotel a group of 240 children had food provision for 50days. After 10days ,40 children left the hotel. For how many days would the remaining food last
The food will therefore be plenty for 60 days. The assumption that the rate of food consumption will remain constant during the provision should be noted as it may not be totally accurate in real-world scenarios.
Mathematically, this can be expressed as:
240 x 50 = C x D
The result of simplifying the left side of the equation is:
12000 = C x D
40 kids left the motel after 10 days, leaving the following number of kids:
240 - 40 = 200
Now, we can calculate how many days the remaining food will endure for 200 kids using the same proportionality equation:
12000 = 200 x D
The result of multiplying both sides of the equation by 200 is:
D = 12000/200 = 60
The food will therefore be plenty for 60 days.
The assumption that the rate of food consumption will remain constant during the provision should be noted as it may not be totally accurate in real-world scenarios.
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HELP!!!! I NEED AN ANSWER QUICK!!! WILL MARK BRAINLIEST!!!!
Please help quick!!!!!
1/3(12x+6x-21) combine like terms
= 1/3(18x-21) distribute
= 6x - 7
Estimate the instantaneous rate of change at x=1
The instantaneous rate of change at x=1.
What is function?In mathematics, a function is a relation between sets that assigns to each element of a given set, exactly one element of a related set. This relationship between two sets is called a mapping, and it is commonly represented using a function notation. Functions are used to describe mathematical relationships and are studied extensively in mathematics, computer science, and other related fields.
The instantaneous rate of change at x=1 can be estimated using the concept of derivatives. A derivative is a mathematical operation that measures the rate at which a function changes with respect to its input. In this case, the derivative of the given function will measure the rate at which the function changes with respect to the input x=1.
The derivative of a function can be calculated using the formula:
f′(x)=limh→0[f(x+h)-f(x)]/h
Where f′(x) is the derivative of the function, f(x) is the function, and h is a small number close to zero.
To calculate the derivative of the function at x=1, we can plug in x=1 into the formula:
f′(1)=limh→0[f(1+h)-f(1)]/h
By calculating this limit, we can estimate the instantaneous rate of change at x=1.
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fencing a garden a homeowner has 100 ft of fencing to enclose a rectangular garden. if the garden is to be 5 ft longer than it is wide, find its dimensions.
The dimensions of the rectangular garden is 25 feet by 50 feet
To find the dimensions of a rectangular garden that is 5 feet longer than it is wide, when the homeowner has 100 feet of fencing to enclose it, you can use the equation for the perimeter of a rectangle (P = 2l + 2w). Since the perimeter of the rectangle is given (100 feet), you can solve for l and w, the length and width of the rectangle.
P = 2l + 2w
100 = 2l + 2w
50 = l + w l = 50 - w
2(50 - w) + 2w = 100
100 - w = 2w , w = 25 l = 50 - 25, l = 25
Therefore, the dimensions of the rectangular garden that is 5 feet longer than it is wide, when the homeowner has 100 feet of fencing to enclose it, is 25 feet by 50 feet.
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Verify the following identities
Answer:
See below for proof.
Step-by-step explanation:
Use the following trigonometric identities to verify the given identities:
[tex]\boxed{\cot(x)=\dfrac{\cos(x)}{\sin(x)}}[/tex]
[tex]\boxed{\sec(x)=\dfrac{1}{\cos(x)} \implies \cos(x)=\dfrac{1}{\sec(x)}}[/tex]
[tex]\boxed{\tan(x)=\dfrac{\sin(x)}{\cos(x)}}[/tex]
[tex]\boxed{\sin^2(x)+\cos^2(x)=1 \implies \cos^2(x)=1-\sin^2(x)}[/tex]
Question a)[tex]\begin{aligned}\dfrac{1}{\sin(x)\cot(x)}&=\dfrac{1}{\sin(x) \cdot\frac{\cos(x)}{\sin(x)}}\\\\&=\dfrac{1}{\frac{\sin(x)\cos(x)}{\sin(x)}}\\\\&=\dfrac{\sin(x)}{\sin(x)\cos(x)}\\\\&=\dfrac{1}{\cos(x)}\end{aligned}[/tex]
Question b)[tex]\begin{aligned}\sec(x)-\tan(x)\sin(x)&=\dfrac{1}{\cos(x)}-\dfrac{\sin(x)}{\cos(x)} \cdot \sin(x)\\\\&=\dfrac{1}{\cos(x)}-\dfrac{\sin^2(x)}{\cos(x)}\\\\&=\dfrac{1-\sin^2(x)}{\cos(x)}\\\\&=\dfrac{\cos^2(x)}{\cos(x)}\\\\&=\cos(x)\\\\&=\dfrac{1}{\sec(x)}\end{aligned}[/tex]
For a particular population, the largest distance (gap) between
a score and the mean is 11 points. The smallest distance between a
score and the mean is 4 points. Therefore, the standard deviation
___
The largest deviation and mean are 11 and the smallest deviation and mean are 4, respectively. Although we cannot give an exact answer, the standard deviation will always fall between 4 and 11. It will be the average of these values.
The standard deviation reveals how far data deviates from the mean. While a high standard deviation indicates that the data are more dispersed, a low standard deviation indicates that the data are clustered around the mean.
Standard deviation is important because it makes measurements easier to understand when the data is distributed.
standardstandard deviation of the data will be higher the more evenly distributed the data is.
The largest deviation and mean are 11 and the smallest deviation and mean are 4, respectively. Although we cannot give an exact answer, the standard deviation will always fall between 4 and 11. It will be the average of these values.
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There are 5,500 milliliters in a bottle of laundry detergent. How many liters is this? 4. 5 liters 5. 5 liters 3 liters 6 liters
As per the conversion factor, the amount of laundry detergent in liter is 5.5 liters (option b).
When we want to convert units from one measurement system to another, we use something called a conversion factor. A conversion factor is a ratio that expresses the relationship between two different units of measurement. In this case, we want to convert milliliters (mL) to liters (L).
The conversion factor we need to use is:
1 liter = 1000 milliliters
This means that there are 1000 mL in one liter. We can use this conversion factor to convert 5,500 mL to liters.
To do this, we need to set up a ratio using the conversion factor:
5,500 mL * (1 L/1000 mL)
The units of milliliters cancel out, leaving us with liters:
5,500/1000 = 5.5 L
So the answer is 5.5 liters.
Hence the correct option is B.
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Ratna wants to pave stone in his triangular yard of sides 50m, 50m and 60m. If the area of each stone is 120cm², find the total number of stones required for it. Also, calculate the total cost at the rate of Rs. 5 per stone
The total number of stones required to pave the yard would be 100,000√3 stones and the cost would be Rs. 500000√3 at the rate of Rs. 5 per stone.
Rate calculationTo find the number of stones required, we need to find the area of the triangular yard and then divide it by the area of each stone.
First, let's find the area of the triangular yard using Heron's formula:
s = (50 + 50 + 60) / 2 = 80
Area = √(s(s-50)(s-50)(s-60)) = √(803030*20) = 1200√3 m²
Now, we need to convert the area to cm² and divide by the area of each stone:
Area in cm² = (1200√3) * 10000 = 12000000√3 cm²
Number of stones required = (Area in cm²) / (Area of each stone) = (12000000√3) / 120 = 100000√3 stones
Finally, to calculate the total cost at the rate of Rs. 5 per stone, we simply multiply the number of stones by the rate per stone:
Total cost = (100000√3) * Rs. 5 = Rs. 500000√3
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x (3x-1) + (3x-1) = 0
Answer:
x = - 1 , x = [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
assuming you require the solution to the equation
x(3x - 1) + (3x - 1) = 0 ← factor out (3x - 1) from each term
(3x - 1)(x + 1) = 0
equate each factor to zero and solve for x
x + 1 = 0 ⇒ x = - 1
3x - 1 = 0 ( add 1 to both sides )
3x = 1 ( divide both sides by 3 )
x = [tex]\frac{1}{3}[/tex]
How do the expressions y=7x3+2 and y=11x3+2 differ?
Answer: A
Step-by-step explanation:
Can someone write this 0.403, 0.04, 0.043, 0.034 in order please?
Answer:
If its from least to greatest, it would be:
0.034, 0.04, 0.043, 0.403
Step-by-step explanation:
Solved using placement of the numbers
Games Galore Super Store buys the latest video game at a wholesale price of $30. 0. The markup rate at Games Galore Super Store is 40%. How much will you pay to purchase a game at the store?
At Games Galore Super Store, you would pay $42 to purchase the latest video game.
The term "purchase" typically refers to the act of buying something, usually with money or some other form of payment. When you make a purchase, you acquire ownership or possession of the item or service that you have bought. Purchases can range from everyday items like food and clothing to larger investments like a car or a house.
If the wholesale price of a video game is $30.00 and the markup rate at Games Galore Super Store is 40%, then the store will sell the game at a price equal to:
$30.00 + (40% * $30.00) = $30.00 + $12.00 = $42.00
Therefore, you would pay $42.00 to purchase a video game at Games Galore Super Store.
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Let \( \alpha \) and \( \beta \) be first quadrant angles with \( \cos (\alpha)=\frac{\sqrt{5}}{10} \) and \( \sin (\beta)=\frac{\sqrt{10}}{10} \). Find \( \cos (\alpha+\beta) \).
The final answer is: cos(α + β) = (3i - √190) / 10.
\( \alpha \) and \( \beta \) are first quadrant angles with \( \cos (\alpha)=\frac{\sqrt{5}}{10} \) and \( \sin (\beta)=\frac{\sqrt{10}}{10} \). We must now compute \( \cos (\alpha+\beta) \).Solution:Given, α and β are the first quadrant angles with:cos(α) = √5 / 10 (1)sin(β) = √10 / 10 (2)We know that sin2θ + cos2θ = 1Also, we know sin(θ1 + θ2) = sinθ1 cosθ2 + cosθ1 sinθ2cos(θ1 + θ2) = cosθ1 cosθ2 - sinθ1 sinθ2Now, we will compute:cos2α = 1 - sin2αcos2α = 1 - [sin(α)]2cos2α = 1 - [1 - cos2α]cos2α = 2cos2α - 1cos2α = (2cos2α - 1) / 2Putting the value of cos(α) from (1), we get:cos2α = (2 × (√5 / 10)2 - 1) / 2cos2α = (5 / 25 - 1) / 2cos2α = -3 / 25cosα = √(-3 / 25)cosα = i√3 / 5Similarly, we can compute the value of sinβ by squaring equation (2).cos2β = 1 - sin2βcos2β = 1 - [sin(β)]2cos2β = 1 - [10 / 100]cos2β = 9 / 10cosβ = √(9 / 10)cosβ = 3√10 / 10Now, we can find the value of sinα from equation (1) by squaring it:cos2α + sin2α = 1sin2α = 1 - cos2αsin2α = 1 - (5 / 100)sin2α = 95 / 100sinα = √(95 / 100)sinα = √19 / 10We have, cos(α + β) = cosα cosβ - sinα sinβNow, we will plug in the values we computed:cos(α + β) = (i√3 / 5) × (3√10 / 10) - (√19 / 10) × (√10 / 10)cos(α + β) = 3i / 5 - √190 / 100cos(α + β) = (3i - √190) / 10The final answer is: cos(α + β) = (3i - √190) / 10.
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Charlotte borrows $9000 to buy a second-hand car.
The loan must be repaid over 5 years at 12% p.a. simple interest. Calculate the:
a total amount to be repaid b monthly repayment amount if the repayments
are spread equally over the 5 years
Answer:
a) To calculate the total amount to be repaid, we need to add the interest to the principal amount.
The formula for simple interest is:
I = P * r * t
where I is the interest, P is the principal amount, r is the interest rate per year, and t is the time in years.
In this case, P = $9000, r = 12% = 0.12, and t = 5 years.
So, the interest on the loan is:
I = $9000 * 0.12 * 5 = $5400
The total amount to be repaid is:
Total amount = Principal + Interest = $9000 + $5400 = $14400
Therefore, the total amount to be repaid is $14,400.
b) To calculate the monthly repayment amount, we need to divide the total amount to be repaid by the number of months in 5 years (60 months), since the repayments are spread equally over 5 years.
Monthly repayment amount = Total amount to be repaid / Number of months
= $14,400 / 60
= $240
Therefore, the monthly repayment amount if the repayments are spread equally over the 5 years is $240.
Use the information in the given table to determine the equation of the function in vertex form.
Answer:
1(4)6/7 37)95/9546(76 is the answer
Answer this question.
According to the question the fraction of the bigger square that is shaded is 50%.
What is fraction?Fraction is a numerical representation of a part of a whole or a ratio between two numbers. It is written in the form of a/b where a is the numerator and b is the denominator. A fraction can refer to a part of a whole number, a part of a set, or a part of a unit. Fractions are commonly used in mathematics, cooking, science, and many other areas.
This is because the ratio of the area of the smaller square to the area of the bigger square to the area of the triangle is 1:3:2. Therefore, the ratio of the area of the shaded parts of the squares to the area of the shaded triangle is 25%: 25%:50%. Since the area of the bigger square is three times that of the smaller square, then 50% of the bigger square must be shaded to maintain the given ratio.
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Nora made 3 gallons of lemonade. Which of the following can be used to find the number of pints of lemonade that Nora made?
Choices:
3x2x3
3x4x2
3x3x2
3x4x3
g a particular telephone number is used to receive both voice calls and fax messages. suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. what is the probability that
The probability is 0.2137
Explanation:
Given a particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls.
The probability that in a sample of 25 incoming calls, exactly 5 of them involve fax messages is calculated as follows;
P(X = 5) = 25C5(0.25)⁵(0.75)²⁰= 53130(0.25)⁵(0.75)²⁰= 0.2137
Therefore, the probability that in a sample of 25 incoming calls, exactly 5 of them involve fax messages is 0.2137.
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In Figure 1, name the relationship between the following
airs of angles.
a. ZBOD and ZDOA
D. ZCOB and
C. ZBOC and ZCOA
lomont of an angle that measures 15°
The relationship between the following pairs of angles.
a. ∠ BOD and ∠ DOA are linear pairs
D. ∠ COB and C. ∠ BOC and ∠ COA are linear pairs
What are linear pairs?Linear pairs are two adjacent angles formed when two straight lines intersect.
The two angles in a linear pair are always supplementary, which means they add up to 180 degrees.
In other words, if one angle in a linear pair measures x degrees, then the other angle measures (180 - x) degrees. Linear pairs are useful in geometry and often used in proofs and problem-solving.
In the problem, the pairs
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An oil company conducts a geological study that indicates that an exploratory oil well should have a 35% chance of striking oil. What is the probability that the second strike comes on the fourth well drilled? Give answer to four decimal places.
0.0329
The question is asking about the probability that the second strike comes on the fourth well drilled. Here, the probability of striking oil is 35%. The probability of not striking oil is 65%. The question is asking about a specific combination of strikes, that is, the second strike comes on the fourth well drilled.Here, we can use the binomial distribution, with the following values, n = 4, p = 0.35, q = 0.65, k = 2Probability of exactly 2 successful attempts out of 4 attempts:The formula for binomial probability is:P(k)= (nck)(p^k)(q^(n−k))where n is the number of trials, k is the number of successful outcomes, p is the probability of success, and q is the probability of failure.The formula for finding the binomial coefficient (nCk) is:nCk = (n!)/(k!(n−k)!)where n is the total number of items, and k is the number of items being chosen.Explanation:Here, n = 4, k = 2, p = 0.35 and q = 0.65So, P(k=2) = (nCk)(p^k)(q^(n−k))= (4C2)(0.35^2)(0.65^(4−2))= (6)(0.1225)(0.4225)= 0.0329So, the probability that the second strike comes on the fourth well drilled is 0.0329 (rounded to four decimal places).
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What is the equation of the line that is parallel to the line y = x 4 and passes through the point (6, 5)? y = x 3 y = x 7 y = 3x – 13 y = 3x 5
The equation of the line that is parallel to the line y = x 4 and passes through the point (6, 5) is y = x - 1
The line y = x + 4 has a slope of 1 because it is in the form y = mx + b, where m is the slope of the line. To find a line parallel to this line, we need to use the same slope of 1.
Now, we can use the point-slope form of a linear equation to find the equation of the line that passes through the point (6, 5) with a slope of 1:
y - y1 = m(x - x1)
where m = 1, x1 = 6, and y1 = 5.
Plugging in the values, we get:
y - 5 = 1(x - 6)
Simplifying, we get:
y - 5 = x - 6
y = x - 1
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4. the sides of a triangle are in the extended ratio 2:6:7. if the perimeter of the triangle is 45 inches, then what is the length of the shortest side?
5. the measures of the angles of a triangle are in the extended ratio 17:16:12. what is the measure of the largest angle?
6. what is the value of X in the proportion? [tex]\frac{2x+7}{3x-4}[/tex]=[tex]\frac{11}{2}[/tex]
Step-by-step explanation:
4. 2/ 15 x 45/1 = 90/15
=6inches
( 15 is the total of the ratio)
5. Sum of angles in a triangle in 180
Total of the ratio is 17+16+12 = 45
Largest angle is 17/ 45 x180/1 = 68 degree
Ans 68degrees
6. 2x + 7/ 3x - 4 = 11/2
Cross multiply
2( 2x + 7) = 11(3x - 4)
Expand the equation
4x +14 = 33x - 44
Subract 4x from both sides
14 = 29x - 44
Add 44 to both sides
58 =29x
Divide both sides by 29
X =2
4. A researcher is studying life expectancy in different parts of the world. Using birth and death
records, she randomly selects a sample of 20 people from Town A and a sample of 20 people
from Town B and records their lifespans in years.
Mean Lifespan in Years Standard Deviation
Town A
Town B
78.5
74.4
11.2
12.3
The researcher wants to test the claim that there is a significant difference in lifespan for people
in the two towns. What is the P-value and conclusion at a significance level of 0.10?
(1 point)
OP-value = 0.277; fail to reject the null hypothesis that the means of the populations are equal
OP-value = 0.076228; fail to reject the null hypothesis that the means of the populations are equal
OP-value = 0.277; reject the null hypothesis that the means of the populations are equal
OP-value = 0.076228; reject the null hypothesis that the means of the populations are equal
The required p-value of 0.277 is greater than the significance level of 0.10, we again fail to reject the null hypothesis.
How to find the P-value?To test the claim of a significant difference in lifespan for people in Town A and Town B, we can use a two-sample t-test. The null hypothesis is that there is no significant difference in the mean lifespan between the two towns, while the alternative hypothesis is that there is a significant difference.
Let [tex]$\mu_A and \mu_B$[/tex] be the true mean lifespans of the populations in Town A and Town B, respectively. Then the null and alternative hypotheses can be written as:
[tex]$H_0: \mu_A = \mu_B$[/tex]
[tex]$H_1: \mu_A \neq \mu_B$[/tex]
We can use the t-test statistic to test this hypothesis, which is calculated as:
[tex]$t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}$[/tex]
where [tex]$\bar{x}_A$[/tex]and [tex]$\bar{x}_B$[/tex] are the sample means, [tex]$s_A$[/tex] and [tex]$s_B$[/tex] are the sample standard deviations, and[tex]$n_A$[/tex] and [tex]$n_B$[/tex] are the sample sizes.
Substituting the given values, we get:
[tex]$t = \frac{78.5 - 74.4}{\sqrt{\frac{11.2^2}{20} + \frac{12.3^2}{20}}} = 1.102$[/tex]
Using a two-tailed t-test with 38 degrees of freedom (20 + 20 - 2), and a significance level of 0.10, the critical value of t is:
[tex]$t_{\alpha/2,38} = \pm 1.686$[/tex]
Since the calculated t-value of 1.102 is less than the critical value of 1.686, we fail to reject the null hypothesis.
To find the p-value, we can use a t-distribution table or a calculator to find the probability of getting a t-value of 1.102 or more extreme (in either direction) if the null hypothesis is true. This is a two-tailed test, so we need to multiply the resulting probability by 2:
p-value = P(|t| > 1.102) = 2P(t > 1.102) = 2(1 - P(t < 1.102)) =0.277
Since the p-value of 0.277 is greater than the significance level of 0.10, we again fail to reject the null hypothesis.
Therefore, at a significance level of 0.10, we do not have sufficient evidence to conclude that there is a significant difference in lifespan for people in Town A and Town B.
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