Parents of teenage boys often complain that auto insurance costs more, on average, for teenage boys than for teenage girls. A group of concerned parents examines a random sample of insurance bills. The mean annual cost for 36 teenage boys was $670. For 23 teenage girls, it was $556. From past years, it is known that the population standard deviation for each group is $180. Determine whether or not you believe that the mean cost for auto insurance for teenage boys is greater than that for teenage girls. Conduct a hypothesis test at the 5% level.NOTE: If you are using a Student's t-distribution for the problem, including for paired data, 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: ?boys > ?girlsH0: ?boys = ?girls H0: ?boys < ?girlsH0: ?boys ≠ ?girlsPart (b)State the alternative hypothesis.Ha: ?boys > ?girlsHa: ?boys ≠ ?girls Ha: ?boys < ?girlsHa: ?boys = ?girlsPart (c)In words, state what your random variable Xboys − Xgirls represents.Xboys − Xgirls represents the average difference in the cost of auto insurance for boys and girls.Xboys − Xgirls represents the difference in the cost of auto insurance for boys and girls. Xboys − Xgirls represents the difference in the average cost of auto insurance for boys and girls.Xboys − Xgirls represents the average cost of auto insurance.Part (d)State the distribution to use for the test. (Round your answers to two decimal places.)Xboys − Xgirls ~ , Part (e)What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer 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 true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $114 less than the sample average cost for girls.If H0 is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $114 more than the sample average cost for girls. If H0 is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $114 more than the sample average cost for girls.If H0 is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $114 less than the sample average cost for girls.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 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 do not reject the null hypothesis.(iv) Conclusion:There is sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls.There is not sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls. Part (i)Explain how you determined which distribution to use.The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples involve the difference in proportions.The t-distribution will be used because the samples are dependent.

Answered: Parents of teenage boys often complain…

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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Parents of teenage boys often complain that auto insurance costs more, on average, for teenage boys than for teenage girls. A group of concerned parents examines a random sample of insurance bills. The mean annual cost for 36 teenage boys was $670. For 23 teenage girls, it was $556. From past years, it is known that the population standard deviation for each group is $180. Determine whether or not you believe that the mean cost for auto insurance for teenage boys is greater than that for teenage girls. Conduct a hypothesis test at the 5% level.

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, 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: ?_boys > ?_girls

H₀: ?_boys = ?_girls

H₀: ?_boys < ?_girls

H₀: ?_boys ≠ ?_girls
Part (b)

State the alternative hypothesis.

H_a: ?_boys > ?_girls

H_a: ?_boys ≠ ?_girls

H_a: ?_boys < ?_girls

H_a: ?_boys = ?_girls
Part (c)

In words, state what your random variable
X_boys − X_girls
represents.

X_boys − X_girls
represents the average difference in the cost of auto insurance for boys and girls.
X_boys − X_girls
represents the difference in the cost of auto insurance for boys and girls.
X_boys − X_girls
represents the difference in the average cost of auto insurance for boys and girls.
X_boys − X_girls
represents the average cost of auto insurance.
Part (d)

State the distribution to use for the test. (Round your answers to two decimal places.)

X_boys − X_girls ~

,
Part (e)

What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer 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 true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $114 less than the sample average cost for girls.If
H₀
is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $114 more than the sample average cost for girls. If
H₀
is true, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is at least $114 more than the sample average cost for girls.If
H₀
is false, then there is a chance equal to the p-value that the sample average annual cost of insurance for boys is $114 less than the sample average cost for girls.
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 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 do not reject the null hypothesis.

(iv) Conclusion:
There is sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls.There is not sufficient evidence to show that the average cost for auto insurance for teenage boys is greater than that for teenage girls.
Part (i)

Explain how you determined which distribution to use.
The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known. The standard normal distribution will be used because the samples involve the difference in proportions.The t-distribution will be used because the samples are dependent.

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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