Hypothesis test for a Normal mean

The weights of bags of rice, X kg, have a normal distribution with unknown mean μ kg and known standard deviation kg. A random sample of 100 bags of rice gave a 90% confidence interval for μ of (0.4633, 0.5127).

Without carrying out any further calculations, use this confidence interval to test whether or not μ = 0.5 State your hypotheses clearly and write down the significance level you have used.

A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg.

Calculate a 95% confidence interval for μ based on this second sample.

Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is 0.5 kg² and the variance of the yield for the new variety of plant is 0.75 kg². A random sample of 60 plants of the old variety has a mean yield of 5.5 kg. A random sample of 70 of the new variety has a mean yield of 7 kg.

Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.

Explain the relevance of the Central Limit Theorem to the test in part (a).

Mr Alan and Ms Burns are two Mathematics teachers teaching mixed ability groups of students in a large college. At the end of the college year all students took the same examination. A random sample of 29 of Mr Alan’s students and a random sample of 26 of Ms Burns’ students are chosen. The results are summarised in the table below.

Stating your hypotheses clearly, test, at the 10% level of significance whether there is evidence that there is a difference in the mean scores of their students. (6)

Ms Burns thinks the comparison was unfair as the examination was set by Mr Alan. She looks up a different set of examination results for these students and, although Mr Alan’s sample has a higher mean, she calculates the test statistic for this new set ofresults to be 1.6 However, Mr Alan now claims that the mean marks of his students are higher than the mean marks of Ms Burns’ students.

Test Mr Alan’s claim, stating the hypotheses and critical values you would use. Use a 10% level of significance. (3)

The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm. The heights of the orchids are normally distributed. Given that the population standard deviation is 0.5 cm,

estimate limits between which 95% of the heights of the orchids lie,

find a 98% confidence interval for the mean height of the orchids.

A grower claims that the mean height of this type of orchid is 19.5 cm.

Comment on the grower’s claim. Give a reason for your answer.

The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm. The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm.

Test, using a 5% level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.

State two assumptions you made in carrying out the test in part (a).