Download Year 12 Normal Distribution Test

Statistics 1Maths Assessed Homework 7: Confidence
intervals
Answer all questions showing your working out clearly
1) The time, X minutes, taken by Fred Fast to install a satellite dish may be
assumed to be a normal random variable with mean 134 and standard
deviation 16.
a) Determine the probability that Fred will take less than 150 minutes to
install a satellite.
b) Calculate P( X< 160)
c) Calculate P( X=136)
d) Determine, to one decimal place, the time exceeded by 10% of
installations.
The time taken by Sid Slow to install a satellite dish may also assumed to
be a normal random variable but with
P( Y< 170) =0.14 and P( Y> 200) = 0.03
e) Determine to the nearest minute the mean and standard deviation of Y.
2) The weights of bags of red gravel may be modeled as a normal
distribution with mean of 25.8 kg and standard deviation 0.5 kg.
a) Determine the probability that a randomly selected bag of red gravel
will weigh less than 25kg.
b) Calculate P( 25.5 < X < 26.5)
c) Determine, to two decimal places, the weight exceeded by 75% of
bags.
3) The weight in grams of Italian grated cheese in cartons may be assumed
to normally distributed with mean, μ, and standard deviation 1.6.
From a random sample of 64 cartons
a) Calculate the standard error.
b) Construct a 95% confidence interval for μ
4) The contents of a random sample of 100 cans of a soft drink are
measured. The results have a mean of 331.28 ml and standard deviation
of 2.97 ml
a)
Show that an unbiased estimate of the population variance is
8.91 ml2
b) Explain why it would be appropriate to use 8.8209 as an
estimate of the variance.
c) Using 8.91 for the variance construct a 99% confidence interval
for the population mean, giving the limits to two decimal places.
d) Explain why, in answering part b) an assumption regarding the
distribution of the cans was not necessary.
5) The weights of the contents of jars of honey may be assumed to be
normally distributed. The weight of the contents of a random sample eight
jars were as follows:
458 450 457 456 460 459 458 456
a) Calculate the mean and standard deviation for the weight of the
sample.
b) Use your answer to part a) to calculate an unbiased estimator
for the variance of the population.
c) Calculate a 95% confidence interval for the mean weight of
the contents of the jars.
d) On each jar it states “ Contents 454 grams”. Comment on this
using the sample and your results for part c)