Percentage of normally distributed apple masses in an interval, two apples both below a mass, and a one-tailed test of whether a fertiliser raises the mean
Percentage of normally distributed apple masses in an interval, two apples both below a mass, and a one-tailed test of whether a fertiliser raises the mean
9 marks
The masses of apples picked in an orchard are normally distributed with mean \(152\) grams and standard deviation \(12\) grams.
- Find the percentage of these apples with a mass between \(140\) grams and \(170\) grams.[2]
- Two apples are picked at random. Find the probability that each of them has a mass of less than \(135\) grams.[3]
- The trees in one part of the orchard are treated with a new fertiliser, and a random sample of \(36\) apples from these trees has a mean mass of \(155.6\) grams. Assume that the standard deviation of the masses is still \(12\) grams. Test, at the \(5\%\) significance level, whether the fertiliser increases the mean mass of the apples. State your null and alternative hypotheses and give a reason for your conclusion.[4]
Solution:
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Answer:(a) \(77.5\%\) (b) \(0.00613\) (c) \(\mathrm{H}_0:\mu=152,\ \mathrm{H}_1:\mu>152\); \(p=0.0359<0.05\), so reject \(\mathrm{H}_0\): the fertiliser increases the mean mass
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