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Hatshy [7]
2 years ago
8

Crash testing is a highly expensive procedure to evaluate the ability of an automobile to withstand a serious accident. A simple

random sample of 12 small cars were subjected to a head-on collision at 40 miles per hour. Of them 8 were "totaled," meaning that the cost of repairs is greater than the value of the car. Another sample of 15 large cars were subjected to the same test, and 5 of them were totaled. Let pX represent the population proportion of small cars that are totaled and let pY represent the population proportion of large cars that are totaled. Find a 95% confidence interval for the difference pX – pY . Round the answers to four decimal places. The 95% confidence interval is ( , ).
Mathematics
1 answer:
Veseljchak [2.6K]2 years ago
6 0

Answer:

The 95% confidence interval is (-0.2451, 06912)

Step-by-step explanation:

From the question, we have;

The number of small cars in the sample of small cars, n₁ = 12

The number of small cars that were totaled, x = 8

The number of large cars in the sample of small cars, n₂ = 15

The number of large cars that were totaled, y = 5

Therefore, the proportion of small cars that were totaled, pX = x/n₁

∴ pX = 8/12 = 2/3

The proportion of large cars that were totaled, pY = y/n₁

∴ pY = 5/15 = 1/3

The 95% confidence interval for the difference pX - pY is given as follows;

pX-pY\pm z^{*}\sqrt{\dfrac{pX\left (1-pX  \right )}{n_{1}}+\dfrac{pY\left (1-pY  \right )}{n_{2}}}

\dfrac{2}{3} -\dfrac{1}{3} \pm 1.96 \times \sqrt{\dfrac{\dfrac{2}{3} \times \left (1-\dfrac{2}{3}   \right )}{12}+\dfrac{\dfrac{1}{3} \times \left (1-\dfrac{1}{3}   \right )}{15}}

Therefore, we have;

\therefore 95\% \  CI = \dfrac{1}{3} \pm 0.3578454

The 95% confidence interval, CI = (-0.2451, 06912)

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6 0
3 years ago
Read 2 more answers
A steamer travels 36 km upstream and 32 km downstream in 6.5 hours. The same steamer travels 4 km upstream and 40 km downstream
rewona [7]

Answer:

The streamer's speed in still water is 90.23 km/h while the stream's speed is 33.33 km/h

Step-by-step explanation:

Let v = streamer's speed in still water and v' = stream's speed. His speed upstream is V = v + v' and his speed downstream is V' = v - v'.

Since he travels 36 km upstream, the time taken is t = 36/V = 36/(v + v').

he travels 32 km downstream, the time taken is t' = 32/V' = 32/(v - v')

The total time is thus t + t' = 36/(v + v') + 32/(v - v')

Since the whole trip takes 6,5 hours,

36/(v + v') + 32/(v - v') = 6.5  (1)

Multiplying each term by (v + v')(v - v'), we have

(v + v')(v - v')36/(v + v') + (v + v')(v - v')32/(v - v') = 6.5(v + v')(v - v')  (1)

(v - v')36 + (v + v')32 = 6.5(v + v')(v - v')  (1)

36v - 36v' + 32v + 32v' = 6.5(v² + v'²)

68v - 4v' = 6.5(v² + v'²)        (2)

Also he travels 4 km upstream, the time taken is t" = 4/V = 4/(v + v').

he travels 40 km downstream, the time taken is t'" = 40/V' = 40/(v - v')

The total time is thus t" + t'" = 4/(v + v') + 40/(v - v')

Since the whole trip takes 180 minutes = 3 hours,

4/(v + v') + 40/(v - v') = 3  (3)

Multiplying each term by (v + v')(v - v'), we have

(v + v')(v - v')4/(v + v') + (v + v')(v - v')40/(v - v') = 3(v + v')(v - v')  (1)

(v - v')4 + (v + v')40 = 3(v + v')(v - v')  (1)

4v - 4v' + 40v + 40v' = 3(v² + v'²)

44v - 36v' = 3(v² + v'²)      (4)

Dividing (2) by (4), we have

(68v - 4v')/(44v - 36v') = 6.5(v² + v'²)/3(v² + v'²)      

(68v - 4v')/(44v - 36v') = 6.5/3

3(68v - 4v') = 6.5(44v - 36v')

204v - 12v' = 286v - 234v'

204v - 286v = 12v' - 234v'

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v = -222v'/82

v = 111v'/41

Substituting v into (2), we have

68v - 4v' = 6.5(v² + v'²)      

68(111v'/41) - 4v' = 6.5[(111v'/41)² + v'²]        

[68(111/41) - 4]v' = 6.5[(111/41)² + 1]v'²    

[68(111/41) - 4]v' = 6.5[(111/41)² + 1]v'²

[7548/41 - 4]v' = 6.5[12321/1681 + 1]v'²

[(7548 - 164)/41]v' = 6.5[(12321 + 1681)/1681]v'²

[7384/41]v' = 6.5[14002/1681]v'²

[7384/41]v' = [91013/1681]v'²

[91013/1681]v'² - [7384/41]v' = 0

([91013/1681]v' - [7384/41])v' = 0

⇒ v' = 0 or ([91013/1681]v' - 7384/41) = 0

⇒ v' = 0 or [91013/1681]v' - 7384/41) = 0

⇒ v' = 0 or v' =  7384/41 × 16841/91013

⇒ v' = 0 or v' =  180.097 × 0.185

⇒ v' = 0 or v' =  33.33 km/h

Since v' ≠ 0, v' = 33.33 km/h

Substituting v' into v = 111v'/41 = 111(33.33 km/h)/41 = 3699.63 km/h ÷ 41 = 90.23 km/h

So, the streamer's speed in still water is 90.23 km/h while the stream's speed is 33.33 km/h

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3 years ago
Which number is greater than -24
Sophie [7]
Hey there!

You are correct! -16 is the right answer! Good job!

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