Stan 和 Berend 都种苹果。他们在同一个农贸市场上

Stan 和 Berend 知道 Berend 样本的样本均值。测试前

3.6.斯坦想通过说些聪明的话来赎回自己，并提议

Maartje 是第三位苹果农民，她决定加入讨论。她

18 .对于 Maartje，您抽取了 20 个样本

Question 1 (23 points)
Stan and Berend both grow apples. They are on the same farmers market and
are making bold claims regarding the sweetness of their apples. As a customer at
the market that witnesses the dispute you decide to help them settle it by testing
their claims. Assume that the sweetness of the apples is normally distributed
for both Berend and Stan.
First you are taking a look at Stan’s apples. Luckily another observant turns
out to be a chemist that has a device that constructs a norm for apple sweetness
that both Stan and Berend consider to be an objective measure. Using the test
you observe a mean sweetness of Stan’s 12 apples of 28.6 and a sample standard
deviation of 6.8.
The sample mean of Berend’s apples was analyzed before you arrived and
Stan and Berend know the sample mean of Berend’s sample. Before testing
whether there is a diﬀerence in the sweetness of the two apple samples, Stan
tells about his visit to his psychic that already told him that the standard
deviation of the sweetness of his apples is 6.8. To get rid of unnecessary noise
he proposes to take the sample standard deviation of 6.8 to be the true standard
deviation.
 Question 1a, 5 points Argue, using the fact that Stan brings up his
psychic story and that he wants to take the standard deviation as given,
whether you can say something about the sample mean of Berend’s apples
compared to the sample mean of Stan’s apples.
You decide to ignore Stan’s ridiculous claims. To annoy Stan you decide to
consider a new sample for Berend. You have drawn a new sample of 18 apples
for Berend with a sample mean of 29.5 and a sample standard deviation of
3.6. Stan wants to redeem himself by saying something smart and proposes to
assume equal variances for the samples to make life easier. Question 1b, 6 points In what sense does the assumption that s = b,
where s and b represent the standard deviation of the apple sweetness
for Stan and Berend respectively, make life easier? Construct a 95 percent
conﬁdence interval under this assumption and argue why your technique
works using this assumption.
Maartje is a third apple farmer and she decides to join the discussion. She
claims that the mean sweetness is not relevant and that it only matters what
percentage of apples reach a sweetness level of at least 30. The fraction of sweet
apples in Stan’s sample using this deﬁnition is denoted by ps = 0:5. In Berend’s
sample this fraction is given by pb = 8
18 . For Maartje you draw a sample of 20
apples and the fraction of sweet apples for her is pm = 0:8.