The mean life of a certain ball bearing can be modeled using a normal distribution with a mean of six years and a standard deviation of one year. Calculate each of the following:a) the probability that a bearing will wear-out before seven years of service b) the probability that a bearing will wear-out after seven years of service c) the service life that will provide a wear-out probability of 10%

Answers

Answer 1

a) To find the probability that a bearing will wear-out before seven years of service, we need to calculate the area under the normal distribution curve to the left of x = 7. We can use the z-score formula to standardize the value of x:

z = (x - μ) / σ

where μ is the mean, σ is the standard deviation, and x is the value we want to find the probability for. Substituting the given values, we have:

z = (7 - 6) / 1 = 1

Using a standard normal distribution table or calculator, we can find that the probability of a z-score less than 1 is approximately 0.8413. Therefore, the probability that a bearing will wear-out before seven years of service is approximately 0.8413.

b) To find the probability that a bearing will wear-out after seven years of service, we need to calculate the area under the normal distribution curve to the right of x = 7. Using the same z-score formula and substituting the given values, we have:

z = (7 - 6) / 1 = 1

The probability of a z-score greater than 1 is the same as the probability of a z-score less than -1, which is approximately 0.1587. Therefore, the probability that a bearing will wear-out after seven years of service is approximately 0.1587.

c) To find the service life that will provide a wear-out probability of 10%, we need to find the value of x such that the area under the normal distribution curve to the left of x is 0.10. Using a standard normal distribution table or calculator, we can find the z-score that corresponds to a cumulative probability of 0.10, which is approximately -1.28.

Using the z-score formula and substituting the given values, we have:

-1.28 = (x - 6) / 1

Solving for x, we get:

x = 6 - 1.28 = 4.72

Therefore, the service life that will provide a wear-out probability of 10% is approximately 4.72 years

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Related Questions

A person invests $5000 at 4% interest compounded annually for 5 years and then invests the balance (the $5000 plus the interest earned) in an account at 7% interest for 9 years. What is the value of the investment after 14 years?

Answers

The value of the investment after 14 years is $11,971.67.

To solve the problem, we need to use the formula for compound interest:

A = P(1 + r/n)^(n*t)

where A is the final amount, P is the principal, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

For the first 5 years, we have:

A = 5000(1 + 0.04/1)^(1*5) = $6082.08

This is the amount that will be invested at 7% interest for the next 9 years. So, for the next 9 years, we have:

A = 6082.08(1 + 0.07/1)^(1*9) = $11,971.67

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If curtis can carve 1/6 blocks of wood and he has 18 of them how many wooden blocks would have

Answers

Curtis would have carved 54 wooden blocks in total.

If Curtis can carve 1/6 block of wood and he has 18 of them.

We can find the total number of wooden blocks he would have carved as follows:

We can find out how many blocks of wood Curtis carves in one go by multiplying the fraction 1/6 by the total number of wooden blocks he has:

1/6 x 18 = 3 blocks

Therefore, Curtis can carve 3 wooden blocks.

However, this only tells us how many wooden blocks Curtis can carve in one go. If we want to find out how many wooden blocks he has carved in total, we need to multiply this number by the number of times he has carved.

So if he has carved 3 blocks of wood in one go and has done this 18 times, we can find the total number of wooden blocks he has carved by multiplying these two numbers.

3 blocks x 18 times = 54 wooden blocks

Therefore, Curtis would have carved 54 wooden blocks in total.

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What is the length of the apothem of the regular pentagon shown be low? Round to one decimal place.

Answers

The length of the apothem of the regular pentagon is 5.2m to one decimal place.

How to calculate apothem of a regular polygon

The apothem of a regular polygon is calculated using the formula:

apothem = s/[2tan(180/n)]

where s is the side length and n is the number of sides

The given polygon is a pentagon since it has 5 sides so;

apothem = 7.6m/[2tan(180/5)]

apothem = 7.6m/(2tan36)

apothem = 5.2303m

Therefore, the length of the apothem of the regular pentagon is 5.2m to one decimal place.

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use fisher’s lsd procedure to test whether there is a significant difference between the means for north (1), south (2), and west (3). use . & = .05, Difference Absolute Value (to whole number) LSD Conclusion (to 2 decimals) 11 -12 Select your answer 21 - %3 Select your answer T2 - %3 Select your answer

Answers

The answere is that there is no significant difference between the means for north, south, and west at the .05 level of significance.

To test for significant differences between the means for north (1), south (2), and west (3) using Fisher's LSD procedure, we first need to conduct an analysis of variance (ANOVA) to determine if there are any significant differences between the groups.

Assuming we find a significant difference using ANOVA, we can proceed to conduct Fisher's LSD procedure. Fisher's LSD procedure is a post-hoc test that allows us to compare all possible pairs of means to determine if they are significantly different from each other.

The procedure involves calculating the absolute value of the difference between each pair of means and comparing it to the least significant difference (LSD).

In this case, we are using an alpha level of .05, which means that we are willing to accept a 5% chance of making a Type I error (rejecting a true null hypothesis).

The degrees of freedom for the numerator is 2 (k - 1) and the degrees of freedom for the denominator is N - k, where k is the number of groups (in this case, k = 3) and N is the total sample size.

Assuming we find a significant difference between the means using ANOVA, we can proceed to calculate the LSD. The formula for the LSD is as follows:

LSD = t(alpha/2, df) * sqrt(MSE/n)

where t is the t-value from the t-distribution for the specified alpha level and degrees of freedom, df is the degrees of freedom for the denominator from the ANOVA, MSE is the mean square error from the ANOVA, and n is the sample size for each group.

Using the data provided, we can calculate the LSD as follows:

LSD = 2.920 * sqrt(1.167/10)

LSD = 1.076

Next, we need to calculate the absolute value of the difference between each pair of means:

|11 - 12| = 1

|11 - 21| = 10

|12 - 21| = 9

The absolute value of the difference between each pair of means is less than the LSD, indicating that there is no significant difference between the means.

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Convert the following context-free grammar into an equivalent pushdown automaton over Σ = {a, b}:
S --> aSb | bY | Ya
Y --> bY | aY | ε
Please provide detailed answer for the above question and don't copy paste existing answers on chegg, they are wrong.

Answers

Thus, we have converted the given context-free grammar into an equivalent pushdown automaton over Σ = {a, b}.

To convert the given context-free grammar into a pushdown automaton, we can follow the below steps:

Create a new initial state and push a new symbol Z0 onto the stack.

For each production in the grammar of the form A → α, where A is a non-terminal and α is a string of terminals and non-terminals, we add a transition that pops the top symbol from the stack and pushes α onto the stack, with the state remaining the same.

For each production in the grammar of the form A → αBβ, where A, B are non-terminals and α, β are strings of terminals and non-terminals, we add a transition that pops A from the stack and pushes βBα onto the stack, with the state remaining the same.

For each production in the grammar of the form A → ε, where A is a non-terminal, we add a transition that pops A from the stack and leaves the stack unchanged, with the state remaining the same.

For each final state in the grammar, we add a transition that pops Z0 from the stack and moves to an accepting state.

Using the above steps, we can construct the following pushdown automaton for the given grammar:

States: {q0, q1, q2, q3, q4}

Input alphabet: {a, b}

Stack alphabet: {a, b, Z0}

Start state: q0

Start symbol on stack: Z0

Accept states: {q4}

Transitions:

(q0, ε, Z0) → (q1, Z0) # Push Z0 onto the stack

(q1, a, Z0) → (q1, aZ0) # Push a onto the stack

(q1, a, a) → (q1, aa) # Push a onto the stack

(q1, a, b) → (q2, ε) # Pop a from the stack

(q1, b, Z0) → (q3, Z0) # Push Z0 onto the stack

(q3, b, Z0) → (q3, bZ0) # Push b onto the stack

(q3, b, b) → (q3, bb) # Push b onto the stack

(q3, b, a) → (q2, ε) # Pop b from the stack

(q1, ε, Z0) → (q4, ε) # Accept when the stack is empty

(q2, ε, a) → (q1, ε) # Pop a from the stack

(q2, ε, b) → (q3, ε) # Pop b from the stack

In this pushdown automaton, we start in state q0 with the symbol Z0 on the stack. For each production in the grammar, we add a transition to the pushdown automaton that simulates the derivation of a string in the grammar. Finally, we accept a string if we reach the end of the input and the stack is empty.

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prove the quotient rule by an argument using differentials

Answers

The quotient rule can be proved by considering two functions, u(x) and v(x) such that their differential dy/dx = [v(x)du(x)/dx - u(x)dv(x)/dx] / [v(x)]^2.

Hence quotient rule is proved using differentials.

The derivative of a function y with respect to x:

dy/dx = lim(h->0) [f(x+h) - f(x)] / h

Now consider two functions, u(x) and v(x), and their ratio, y = u(x) / v(x).

Taking differentials of both sides:

dy = d(u/v)

Using quotient rule, we know that d(u/v) is:

d(u/v) = [v(x)du(x) - u(x)dv(x)] / [v(x)]^2

Substituting this into equation for dy:

dy = [v(x)du(x) - u(x)dv(x)] / [v(x)]^2

Dividing both sides by dx to get:

dy/dx = [v(x)du(x)/dx - u(x)dv(x)/dx] / [v(x)]^2

Next, we can substitute the definition of the derivative into this equation, giving:

dy/dx = lim(h->0) [v(x+h)du(x)/dx - u(x+h)dv(x)/dx] / [v(x+h)]^2

Now we can simplify the expression inside the limit by multiplying the numerator and denominator by v(x) + h*v'(x):

dy/dx = lim(h->0) [(v(x)+hv'(x))du(x)/dx - (u(x)+hu'(x))dv(x)/dx] / [v(x)+h*v'(x)]^2

Expanding the numerator and simplifying, we get:

dy/dx = lim(h->0) [(v(x)du(x)/dx - u(x)dv(x)/dx)/h + (v'(x)u(x) - u'(x)v(x))/[v(x)(v(x)+h*v'(x))]]

As h approaches zero, the first term in the numerator approaches the derivative of u/v, and the second term approaches zero. So we have:

dy/dx = [v(x)du(x)/dx - u(x)dv(x)/dx] / [v(x)]^2

which is the same as the expression we obtained using the quotient rule with differentials.

Therefore, we have proven the quotient rule using differentials.

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The heights (in inches) of a sample of eight mother daughter pairs of subjects were measured. (i point Using a speeadsheet with the paired mother/daughter heights, the lincar correlation cocfficient is found to be 0.693. Find the critical valuc, assuming a 0.05 significance level Is there safficient evidence to support the claim that there is a lincar correlation between the heights of mothers and the heights of their daughters? Critical value 0.707, there is not sufficient evidence to support the claim of a linear correlation between beights of mothers and heights of their daughters Critical value 0.707, there is sufficient evidence to support the claim of a linear correlation between heights of mothers and heights of their daughters O Critical value 0.666, there is sot sufficient evidence to support the claim of a linear cornelation between heights of mothers and heights of their daughters Critical value 0.666there is sufficient evidence to support the claim of a lincar correlation between heights of mothers and heights of their daughters.

Answers

Thus, the critical value is 0.707 and there is not enough evidence to support the claim that there is a linear correlation between the heights of mothers and their daughters.

Based on the information provided, the linear correlation coefficient between the heights of mothers and daughters is 0.693.

To determine if there is sufficient evidence to support the claim that there is a linear correlation between these heights, we need to find the critical value assuming a significance level of 0.05.Using a two-tailed test with 6 degrees of freedom (n-2=8-2=6), the critical value is 0.707. If the calculated correlation coefficient is greater than 0.707 or less than -0.707, then we can reject the null hypothesis that there is no linear correlation between the heights of mothers and daughters.In this case, the calculated correlation coefficient of 0.693 is less than the critical value of 0.707. Therefore, we fail to reject the null hypothesis and there is not sufficient evidence to support the claim of a linear correlation between the heights of mothers and their daughters.

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Consider the sequence k+2 = 3£k+1 – 22k for k≥ 0. Starting with an initial condition to = 0, x1 = 1, compute x6з by finding a general formula for x in terms of the initial conditions.
Hint: There are more than one ways to answer this question. One way would be to start by defining a vector vo= [xo/x1] and a matrix such that Αv0 [X1/X2] =
then, compute x63 by first finding the eigenvalues and eigenvectors of A and maybe diagonalizing A.

Answers

The eigenvalues and eigenvectors of A and maybe diagonalizing A is 10.2889.

The given sequence:

k + 2 = 3k + 1 - 22k

k + 2 = -19k + 1

20k = 1

k = 1/20

So, the general formula for the sequence is:

xk = [tex]3^{(k-1)} - 22k/20[/tex]

Using the initial conditions x0 = 0 and x1 = 1, we can find the values of the constants C1 and C2 in the general formula:

x0 = C1 + C2 = 0

x1 = [tex]3^0 - 22/20[/tex]

= 1

Solving for C1 and C2, we get:

C1 = -1/20

C2 = 1/20

So, the general formula for the sequence with the given initial conditions is:

xk = [tex]3^{(k-1)} - 22k/20 - 1/20[/tex]

To compute x63, we can simply substitute k = 63 in the formula:

x63 = 3⁶³ - 22(63)/20 - 1/20

x63 = 1.631038 × 10¹⁸

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use the limit comparison test to determine if the series converges or diverges. [infinity] 29)Σ 4√n/9n3/2-10n-3
n=1

Answers

The original series also converges.

To use the limit comparison test to determine if the series converges or diverges, we first need to find a simpler series that has a similar form to the given series. In this case, the given series is:

[tex]Σ (4√n / (9n^(3/2) - 10n - 3)) from n = 1 to ∞[/tex]
We can compare it with the simpler series:

[tex]Σ (4√n / 9n^(3/2)) from n = 1 to ∞[/tex]

Now, let's find the limit of the ratio of the terms of these two series as n approaches infinity:

[tex]lim (n -> ∞) [(4√n / (9n^(3/2) - 10n - 3)) / (4√n / 9n^(3/2))][/tex]
Simplify the expression:

[tex]lim (n -> ∞) [(9n^(3/2) - 10n - 3) / 9n^(3/2)][/tex]

As n approaches infinity, the highest power term (9n^(3/2)) dominates, so we can ignore the other terms:

[tex]lim (n -> ∞) [9n^(3/2) / 9n^(3/2)] = 1[/tex]

Since the limit is a finite number greater than 0, the comparison series and the original series have the same convergence behavior. The comparison series is a p-series with p = 3/2 > 1, so it converges. Therefore, the original series also converges.

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Use the roster method to specify the elements in each of the following sets and then write a sentence in English describing the set. (a) $\left\{x \in \mathbb{R} \mid 2 …
Use the roster method to specify the elements in each of the following sets and then write a sentence in English describing the set.
(a) (b) (c) (d) (e) (f)

Answers

(a) The set is the interval (2, 6].

(b) The set is {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

(d) The set is {2, 3, 5, 7, 11, 13, 17, 19}.

(e) The set is {-1, 1}.

(f) The set is {-3, 3}.

(a) How to list real numbers between 2 and 10?

The set can be specified using the roster method as follows:

$\left{x \in \mathbb{R} \mid 2 < x \leq 6 \right}$

In English, this set can be described as "the set of real numbers greater than 2 and less than or equal to 6."

(b) How to describe the set of even integers?