There are a couple of questions I still can't figure out after the paper. I can't get Q5 part (ii) (are we supposed to use the graph?) but I'm typing out the whole question. Q10 is simple enough except for the 'combining the two samples' part - I'm sure it's very logical but we've never attempted such a question before. Can anyone help? Thank you.
Qn 5) A spot of light on a computer screen moves in a horizontal line across the screen. At time t seconds, its distance, x mm, from the left-hand edge of the screen is give, for t >= 0, by
x = t^3 - 12t^2 + kt,
where k is a positive constant. Find the set of values of k for which x is an increasing function of t.
It is now give that k = 36.
(i) Sketch the graph of x against t.
(ii) The screen has width 375mm. Find the time in seconds at which the spot reaches the right-hand edge of the screen, giving your answer correct to 1 decimal place.
Qn 10) A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of 150 hours. A random sample of 70 batteries of this type is tested and the lifetime, x hours, of each battery is measured. The results are summarised by
Σx= 10 317, Σx^2= 1 540 231
The population mean lifetime is denoted by μ hours. The null hypothesis μ = 150 is to be tested against the alternative hypothesis μ < 150. Find the p-value of the test and state the meaning of this p-value in the context of the question.
A second random sample of 50 batteries of this type is tested and the lifetime, y hours, of each battery is measured, with results summarised by
Σy= 7331, Σy^2= 1 100 565
Combining the two samples into a single sample, carry out a test, at the 10% significance level, of the same null and alternative hypothesis.
Originally posted by Branwen:Qn 5) A spot of light on a computer screen moves in a horizontal line across the screen. At time t seconds, its distance, x mm, from the left-hand edge of the screen is give, for t >= 0, by
x = t^3 - 12t^2 + kt,
where k is a positive constant. Find the set of values of k for which x is an increasing function of t.It is now give that k = 36.
(i) Sketch the graph of x against t.
(ii) The screen has width 375mm. Find the time in seconds at which the spot reaches the right-hand edge of the screen, giving your answer correct to 1 decimal place.
(i)
x = t^3 - 12t^2 + kt
dx/dt = 3t^2 - 24t + k
For x to be increasing function of t, dx/dt > 0?
[Need to get this verified]
3t^2 - 24t + k > 0
k > 24t - 3t^2
(ii)
x = t^3 - 12t^2 + 36t
375 = t^3 - 12t^2 + 36t
Using GC,
t=11.6689
t=11.7s (1dp)
Originally posted by Branwen:Qn 10) A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of 150 hours. A random sample of 70 batteries of this type is tested and the lifetime, x hours, of each battery is measured. The results are summarised by
Σx= 10 317, Σx^2= 1 540 231The population mean lifetime is denoted by μ hours. The null hypothesis μ = 150 is to be tested against the alternative hypothesis μ < 150. Find the p-value of the test and state the meaning of this p-value in the context of the question.
A second random sample of 50 batteries of this type is tested and the lifetime, y hours, of each battery is measured, with results summarised by
Σy= 7331, Σy^2= 1 100 565
Combining the two samples into a single sample, carry out a test, at the 10% significance level, of the same null and alternative hypothesis.
Unbiased estimate of mean of new sample
= (10317 + 7331) / 120
= 2206/15
= 147.0667
Unbiased estimate of variance of new sample
= (1/119) [1540231+1100565 - (10317+7331)^2 / 120]
= 381.2056
Conduct z-test.
p-value = 0.049904 < 0.10
Therefore reject H0, conclude that there is sufficient evidence to show that the mean lifetime of the batteries is less than 150 hours.
I think is somewhat like this, I just merely combined the 2 samples.
H1 Maths only has 1 paper? No more?
Yep that's all.
And I think you're correct for all the questions, thank you!
Just wondering, how do you obtain the answer for 5(ii) with the GC?
I use the graph method, graph it out then look for the roots.
Did you get the answer key or what, how'd you know I'm right.
Er I said I think. Haha because it's logical and what I would have done. Except that I forgot you could use just use intersects to find t!
Thank you again.
Originally posted by secretliker:(i)
x = t^3 - 12t^2 + kt
dx/dt = 3t^2 - 24t + k
For x to be increasing function of t, dx/dt > 0?[Need to get this verified]
3t^2 - 24t + k > 0
(-24)^2 - 4(3)(k)<0
576-12k<0
12k>576
k> 48