Sampling distributions

Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.

Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar.

A random sample of 5 medium bags of sugar is taken.

Find the value of d so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than d grams.

A restaurant states that its hamburgers contain 20% fat. Paul claims that the mean fat content of their hamburgers is less than 20%. Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5% with a standard deviation of 1.5% You may assume that the fat content of hamburgers is normally distributed.

Find the 90% confidence interval for the mean fat content of hamburgers from the restaurant.

State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers.

The restaurant changes the mean fat content of their hamburgers to % and adjusts the standard deviation to 2%. Paul takes a sample of size n from this new batch of hamburgers. He uses the sample mean X – as an estimator of .

Find the minimum value of n such that P(X – 0.5) 0.9

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A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3. Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than 5%.

The continuous random variable X is uniformly distributed over the interval [ a – 1, a + 5] where a is a constant. Fifty observations of X are taken, giving a sample mean of 17.2

Use the Central Limit Theorem to find an approximate distribution for

Hence find a 95% confidence interval for a.

Explain what you understand by the Central Limit Theorem. (2) A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, x miles, of the brake it has replaced. The garage takes a random sample of 100 brake pads and finds that

Find a 95% confidence interval for the mean lifetime of a brake pad. (5)

Explain the relevance of the Central Limit Theorem in part (b). (2) Brake pads are made to be changed every 20000 miles on average. The hire car company complain that the garage is changing the brake pads too soon.

Comment on the hire company’s complaint. Give a reason for your answer. (2)

A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.

A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below. 120.3 120.1 120.4 120.2 119.9

Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.

The lengths of climbing rope are known to have a standard deviation of 0.2 m. The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of n, lies within 0.05 m of its true value.

Find the minimum sample size required.