A consumer agency is investigating the blowout pressures of Soap Stone tires. A Soap Stone tire is said to blow out when it separates from the wheel rim due to impact forces usually caused by hitting a rock or a pothole in the road. A random sample of 28 Soap Stone tires were inflated to the recommended pressure, and then forces measured in foot-pounds were applied to each tire (1 foot-pound is the force of 1 pound dropped from a height of 1 foot). The customer complaint is that some Soap Stone tires blow out under small-impact forces, while other tires seem to be well made and don't have this fault. For the 28 test tires, the sample standard deviation of blowout forces was 1358 foot-pounds. Soap Stone claims its tires will blow out at an average pressure of 24,000 foot-pounds, with a standard deviation of 1020 foot-pounds. The average blowout force is not in question, but the variability of blowout forces is in question. Using a 0.01 level of significance, test the claim that the variance of blowout pressures is more than Soap Stone claims it is.(i) Give the value of the level of significance. State the null and alternate hypotheses.H0: σ2 = 1,040,400; H1: σ2 > 1,040,400H0: σ2 = 1,040,400; H1: σ2 < 1,040,400 H0: σ2 = 1,040,400; H1: σ2 ≠ 1,040,400H0: σ2 < 1,040,400; H1: σ2 = 1,040,400 (ii) Find the sample test statistic. (Round your answer to two decimal places.) (iii) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.) (iv) Conclude the test.Since P-value ≥ α, we fail to reject the null hypothesis.Since P-value < α, we reject the null hypothesis. Since P-value < α, we fail to reject the null hypothesis.Since P-value ≥ α, we reject the null hypothesis. (v) Interpret the conclusion in the context of the application.At the 1% level of significance, there is insufficient evidence to conclude that the variance is greater than claimed.At the 1% level of significance, there is sufficient evidence to conclude that the variance is greater than claimed.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
A consumer agency is investigating the blowout pressures of Soap Stone tires. A Soap Stone tire is said to blow out when it separates from the wheel rim due to impact forces usually caused by hitting a rock or a pothole in the road. A random sample of 28 Soap Stone tires were inflated to the recommended pressure, and then forces measured in foot-pounds were applied to each tire (1 foot-pound is the force of 1 pound dropped from a height of 1 foot). The customer complaint is that some Soap Stone tires blow out under small-impact forces, while other tires seem to be well made and don't have this fault. For the 28 test tires, the sample standard deviation of blowout forces was 1358 foot-pounds.
Soap Stone claims its tires will blow out at an average pressure of 24,000 foot-pounds, with a standard deviation of 1020 foot-pounds. The average blowout force is not in question, but the variability of blowout forces is in question. Using a 0.01 level of significance, test the claim that the variance of blowout pressures is more than Soap Stone claims it is.
(i) Give the value of the level of significance.
State the null and alternate hypotheses.
H0: σ2 = 1,040,400; H1: σ2 > 1,040,400
H0: σ2 = 1,040,400; H1: σ2 < 1,040,400
H0: σ2 = 1,040,400; H1: σ2 ≠ 1,040,400
H0: σ2 < 1,040,400; H1: σ2 = 1,040,400
(ii) Find the sample test statistic. (Round your answer to two decimal places.)
(iii) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)
(iv) Conclude the test.
Since P-value ≥ α, we fail to reject the null hypothesis.
Since P-value < α, we reject the null hypothesis.
Since P-value < α, we fail to reject the null hypothesis.
Since P-value ≥ α, we reject the null hypothesis.
(v) Interpret the conclusion in the context of the application.
At the 1% level of significance, there is insufficient evidence to conclude that the variance is greater than claimed.
At the 1% level of significance, there is sufficient evidence to conclude that the variance is greater than claimed.
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