MATHEMATICS COLLEGE

"The municipal transportation authority determined that 58% of all drivers were speeding along a busy street. In an attempt to reduce this percentage, the city put up electronic speed monitors so that drivers would be warned if they were driving over the speed limit. A follow-up study is now planned to see if the speed monitors were effective. The null and alternative hypotheses of the test are H0 : π = 0.58 versus Ha : π < 0.58. It is planned to use a sample of 150 drivers taken at random times of the week and the test will be conducted at the 5% significance level. (a) What is the most number of drivers in the sample that can be speeding and still have them conclude that the alternative hypothesis is true? (b) Suppose the true value of π is 0.52. What is the power of the test? (c) How could the researchers modify the test in order to increase its power without increasing the probability of a Type I Error?"

Answers

Answer 1

Answer:

Answer:

a) X=77 drivers

b) Power of the test = 0.404

c) Increasing the sample size.

Step-by-step explanation:

This is a hypothesis test of proportions. As the claim is that the speed monitors were effective in reducing the speeding, this is a left-tail test.

For a left-tail test at a 5% significance level, we have a critical value of z that is zc=-1.645. This value is the limit of the rejection region. That means that if the test statistic z is smaller than zc=-1.645, the null hypothesis is rejected.

The proportion that would have a test statistic equal to this critical value can be expressed as:

$p_c=\pi+z_c\cdot\sigma_p$

The standard error of the proportion is:

$\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.58*0.42)/(150)}\n\n\n \sigma_p=√(0.001624)=0.04$

Then, the proportion is:

$p_c=\pi+z_c\cdot\sigma_p=0.58-1.645*0.04=0.58-0.0658=0.5142$

This proportion, with a sample size of n=150, correspond to

$x=n\cdot p=150\cdot0.5142=77.13\approx 77$

The power of the test is the probability of correctly rejecting the null hypothesis.

The true proportion is 0.52, but we don't know at the time of the test, so the critical value to make a decision about rejecting the null hypothesis is still zc=-1.645 corresponding to a critical proportion of 0.51.

Then, we can say that the probability of rejecting the null hypothesis is still the probability of getting a sample of size n=150 with a proportion of 0.51 or smaller, but within a population with a proportion of 0.52.

The standard error has to be re-calculated for the new true proportion:

$\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.52*0.48)/(150)}\n\n\n \sigma_p=√(0.001664)=0.041$

Then, we calculate the z-value for this proportion with the true proportion:

$z=(p-\pi')/(\sigma_p)=(0.51-0.52)/(0.041)=(-0.01)/(0.041)=-0.244$

The probability of getting a sample of size n=150 with a proportion of 0.51 or lower is:

$P(p<0.51)=P(z<-0.244)=0.404$

Then, the power of the test is β=0.404.

The only variable left to change in the test in order to increase the power of the test is the sample size, as the significance level can not be changed (it is related to the probability of a Type I error).

It the sample size is increased, the standard error of the proprotion decreases. As the standard error tends to zero, the critical proportion tend to 0.58, as we can see in its equation:

$\lim_(\sigma_p \to 0) p_c=\pi+ \lim_(\sigma_p \to 0)(z_c\cdot\sigma_p)=\pi=0.58$

Then, if the critical proportion increases, the z-score increases, and also the probability of rejecting the null hypothesis.

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Heavy children: Are children heavier now than they were in the past? The National Health and Nutrition Examination Survey (NHANES) taken between 1999 and 2002 reported that the mean weight of six-year-old girls in the United States was 49.3 pounds. Another NHANES survey, published in 2008, reported that a sample of 190 six-year-old girls weighed between 2003 and 2006 had an average weight of 46 pounds. Assume the population standard deviation is =σ17 pounds. Can you conclude that the mean weight of six-year-old girls in 2006 is different from what it was in 2002? Use the =α0.10 level of significance and the critical value method.
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Coach Johnson has 450 baseballs to give to 30 players. How many baseballs will each player get?

Answers

Answer:

15. Hope it helps!

Step-by-step explanation:

PLEASE ANSWER I NEED ITTY

Answers

Answer:

13. B

14. A

both declining slopes

Stanley Bank offers a money market account that earns 1.85% compounded continuously: a) If $10,000 is invested in this type of account, how much will it be worth in 3 years? b) How long will it take for the account to be worth $15,000?

Answers

Answer:

a) $1056.33 b) 23 years

Step-by-step explanation:

a) 10000(1+1.85/100)^3=10565.33 (2d.p.)

b) let x be the no. of years

15000 = 10000(1+1.85/100)^x

1.5 = 1.0185^x

ln both sides

ln 1.5 = x ln 1.0185

x = ln 1.5/ln 1.0185

=22.11

=23 years (rounded up)

Final answer:

To calculate the future value of an investment using continuous compounding, you use the formula A = P*e^(rt). For a $10,000 investment at 1.85% annual interest, its value after 3 years is calculated by substituting the given values into the formula. To find out how many years it will take for the investment to reach $15,000, rearrange the formula to solve for t: t = ln(A/P)/r, and substitute the values.

Explanation:

The subject of your question is related to the mathematical concept known as continuouscompound interest, which Stanley Bank is applying to its money market account. In this concept, the formula is A = P×e^(rt), where A is the final amount that will be accumulated after t years, P is the principal amount or the initial investment, r is the interest rate in decimals (1.85% would be 0.0185), and e is Euler's number (~2.72).

a) To calculate the value of an investment of $10,000 after 3 years with an annual interest rate of 1.85%, you would use the formula: A = $10,000 × e^(0.0185 ×3). This will give you the total value of the investment after 3 years.

b) To calculate the number of years it will take for your investment to amount to $15,000 with the same interest rate, you would need to re-arrange the formula to solve for t: t = ln(A/P) / r. So, it would be calculated as: t = ln($15,000/$10,000) / 0.0185. This would give you the number of years it will take for your initial investment to reach $15,000.

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You have been saving dimes for long time and take all of them to the bank to exchange for dollar bills. the teller weighs the dimes and records a total mass of 3358 g. of a dime has an average mass of 2:30g. how much money are you owed? chemistry

Answers

Since 2:30 gm so 1 dime is 15 grams . So if we divide the total mass/ individual mass = 3358/15 =223.866 approx 224 dimes.

Six college buddies bought each other Christmas gifts. They spent:________.1. $178.622. $247.583. $228.454. $176.645. $180.226. $268.45What was the mean amount spent? Round answer to the nearest cent.a. $243.96b. $213.30c. $319.95d. $255.96

Answers

Answer:

B. $213.30

Step-by-step explanation:

Mean amount spent on Christmas gifts = Σx / n

Where,

Σ= sum of

x= cost of each Christmas gifts

n= number of Christmas gift

Mean amount spent on Christmas gifts = Σx / n

= ( $178.622 + $247.583 + $228.454 + $176.645 + $180.226 + $268.45 ) / 6

= $1,279.98 / 6

= $213.33

Round to the nearest cent

= $213.30

Option b is the correct answer

Final answer:

The mean amount spent by six college buddies on Christmas gifts, They spent: approximately $213.33 when rounded to the nearest cent.

Explanation:

The process is quite straightforward and involves the principles of statistics, particularly the calculation of the arithmetic mean. Here are the steps we can follow to solve this problem:

First, we need to find the total amount spent. We do this by adding the amounts together: $178.62 + $247.58 + $228.45 + $176.64 + $180.22 + $268.45. When we add these values, we get a total of $1,279.96.
Next, we need to calculate the mean, which in statistics is also commonly referred to as the average. Since there were six buddies, we divide the total amount ($1,279.96) by 6. When we do this, we find that the mean or average amount spent is approximately $213.33 (when rounded to the nearest cent).

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What is the value of the expression? 8 + 2 x 3--------------
5 x 2 - 8

Answers

2 x 3 6

-------------- = -------- = 3

5 x 2 - 8 2

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