An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 40 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% significance level?Part (a)State the null hypothesis.H0: ? < 4.5H0: ? ≠ 4.5 H0: ? ≥ 4.5H0: ? = 4.5Part (b)State the alternative hypothesis.Ha: ? > 4.5Ha: ? ≠ 4.5 Ha: ? ≥ 4.5Ha: ? = 4.5Part (c)In words, state what your random variable X represents.X represents the average number of students who receive an undergraduate degree in California.X represents the length of time it takes a student to finish his or her undergraduate degree. X represents the average number of students in the California university system.X represents the average length of time it takes students to finish their undergraduate degrees.Part (d)State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.) Part (e)What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.) = Part (f)What is the p-value? (Round your answer to four decimal places.)Explain what the p-value means for this problem.If H0 is false, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more.If H0 is true, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more. If H0 is true, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more.If H0 is false, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more.Part (g)Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h)Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)? = (ii) Decision:reject the null hypothesisdo not reject the null hypothesis (iii) Reason for decision:Since p-value < ?, we do not reject the null hypothesis.Since p-value > ?, we do not reject the null hypothesis. Since p-value > ?, we reject the null hypothesis.Since p-value < ?, we reject the null hypothesis.(iv) Conclusion:There is sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years.There is not sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years. Part (i)Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to two decimal places.)

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Answered: An article in the San Jose Mercury News…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 40 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% significance level?

Part (a)

State the null hypothesis.

H₀:
? < 4.5
H₀:
? ≠ 4.5
H₀:
? ≥ 4.5
H₀:
? = 4.5
Part (b)

State the alternative hypothesis.

H_a:
? > 4.5
H_a:
? ≠ 4.5
H_a:
? ≥ 4.5
H_a:
? = 4.5
Part (c)

In words, state what your random variable X represents.

X
represents the average number of students who receive an undergraduate degree in California.
X
represents the length of time it takes a student to finish his or her undergraduate degree.
X
represents the average number of students in the California university system.
X
represents the average length of time it takes students to finish their undergraduate degrees.
Part (d)

State the distribution to use for the test. (Enter your answer in the form z or t_df where df is the degrees of freedom.)
Part (e)

What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.)
=
Part (f)

What is the p-value? (Round your answer to four decimal places.)

Explain what the p-value means for this problem.
If
H₀
is false, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more.If
H₀
is true, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more. If
H₀
is true, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more.If
H₀
is false, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more.
Part (g)

Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.
Part (h)

Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.
(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
? =

(ii) Decision:
reject the null hypothesisdo not reject the null hypothesis

(iii) Reason for decision:
Since p-value < ?, we do not reject the null hypothesis.Since p-value > ?, we do not reject the null hypothesis.    Since p-value > ?, we reject the null hypothesis.Since p-value < ?, we reject the null hypothesis.

(iv) Conclusion:
There is sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years.There is not sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years.
Part (i)

Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to two decimal places.)

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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