1955, Life Magazine reported that a 25-year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 70 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was 10. Does it appear that the mean work week has increased for women at the 5% level?Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)Part (a)State the null hypothesis.H0:μ ≠ 80H0:μ ≥ 80 H0:μ ≤ 80H0:μ < 80Part (b)State the alternative hypothesis.Ha:μ ≥ 80Ha:μ = 80 Ha:μ > 80Ha:μ ≠ 80Part (c)In words, state what your random variable X represents.Xrepresents the number of hours a woman works in a week.Xrepresents the number of women who work over 80 hours a week. Xrepresents the average number of hours women work in one week.Xrepresents the average number of women who work over 80 hours a week.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.) ---Select--- t z =Part (f)What is the p-value?p-value < 0.0100.010 < p-value < 0.050 0.050 < p-value < 0.100p-value > 0.100Explain what the p-value means for this problem.IfH0is true, then there is a chance equal to the p-value that the average number of hours women work each week is 83 hours or more.IfH0is true, then there is a chance equal to the p-value that the average number of hours women work each week is not 83 hours or more. IfH0is false, then there is a chance equal to the p-value that the average number of hours women work each week is not 83 hours or more.IfH0is false, then there is a chance equal to the p-value that the average number of hours women work each week is 83 hours. or morePart (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 reject the null hypothesis. Since ? > p-value, we do not reject the null hypothesis.Since ? < p-value, we reject the null hypothesis.(iv) Conclusion:There is sufficient evidence to conclude that the average number of hours women work each week is more than 80 hours.There is not sufficient evidence to conclude that the average number of hours women work each week is more than 80 hours. Part (i)Construct a 95% confidence interval for the true mean. Sketch the graph of the situation.

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1955, Life Magazine reported that a 25-year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 70 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was 10. Does it appear that the mean work week has increased for women at the 5% level? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. H0: μ ≠ 80 H0: μ ≥ 80 H0: μ ≤ 80 H0: μ < 80 Part (b) State the alternative hypothesis. Ha: μ ≥ 80 Ha: μ = 80 Ha: μ > 80 Ha: μ ≠ 80 Part (c) In words, state what your random variable X represents. X represents the number of hours a woman works in a week. X represents the number of women who work over 80 hours a week. X represents the average number of hours women work in one week. X represents the average number of women who work over 80 hours a week.

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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1955, Life Magazine reported that a 25-year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 70 women were surveyed with the following results. The sample mean was 83; the sample standard deviation was 10. Does it appear that the mean work week has increased for women at the 5% level?

Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Part (a)

State the null hypothesis.

H₀:
μ ≠ 80
H₀:
μ ≥ 80
H₀:
μ ≤ 80
H₀:
μ < 80
Part (b)

State the alternative hypothesis.

H_a:
μ ≥ 80
H_a:
μ = 80
H_a:
μ > 80
H_a:
μ ≠ 80
Part (c)

In words, state what your random variable X represents.

X
represents the number of hours a woman works in a week.
X
represents the number of women who work over 80 hours a week.
X
represents the average number of hours women work in one week.
X
represents the average number of women who work over 80 hours a week.
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.)
---Select--- t z =
Part (f)

What is the p-value?

p-value < 0.0100.010 < p-value < 0.050 0.050 < p-value < 0.100p-value > 0.100

Explain what the p-value means for this problem.
If
H₀
is true, then there is a chance equal to the p-value that the average number of hours women work each week is 83 hours or more.If
H₀
is true, then there is a chance equal to the p-value that the average number of hours women work each week is not 83 hours or more. If
H₀
is false, then there is a chance equal to the p-value that the average number of hours women work each week is not 83 hours or more.If
H₀
is false, then there is a chance equal to the p-value that the average number of hours women work each week is 83 hours. 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 reject the null hypothesis.     Since ? > p-value, we do not reject the null hypothesis.Since ? < p-value, we reject the null hypothesis.

(iv) Conclusion:
There is sufficient evidence to conclude that the average number of hours women work each week is more than 80 hours.There is not sufficient evidence to conclude that the average number of hours women work each week is more than 80 hours.
Part (i)

Construct a 95% confidence interval for the true mean. Sketch the graph of the situation.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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