Question

A gardener plants 300 sunflower seeds (of a brand called KwikGrow) and, after 2 weeks, measures the seedlings’ heights (in mm). These heights are recorded below: height 38.6 39.1 50.4 49.2 46.2 0 43.1 49.6 16.1 17.9 11.6 50.2 36.5 50.6 40.3 36.3 14.3 40.8 12.1 43.7 47.2 49.9 37.9 49.1 53 47.7 13.8 38.3 49.2 50.6 49.6 52.3 19.8 12.8 46.9 35.3 38.7 39.3 12.4 51.9 36.9 20.8 51.7 38.8 41.9 18.4 41.4 48.7 16.3 50 13.8 50.3 47.6 42 14.9 41.1 43.7 10.1 36.4 40.1 14.9 50.3 12.3 44.3 49.1 10.7 14.9 48.2 14.8 38 41.4 39.4 11.9 13.8 0 35.1 37.3 47.5 12.5 11.8 16 15.7 38.1 58.6 51.2 37.4 36.4 40.8 35.2 15.5 55.9 42.1 47.9 41.1 38.5 51.1 41 53.8 41.5 38.6 48 50.4 17.9 50.8 39.2 13.5 35.9 12.4 16.5 47.9 38.4 11.7 49.4 44.7 45.8 14 40.4 48.9 44.6 17.6 12.4 43.1 18.3 20 17.1 50.1 57.6 50.9 50.2 18.3 0 14.5 40 49.3 51.9 16.1 47 14.6 48.7 38.1 39.7 39.3 0 37.1 13 17.4 37.7 41.3 39.6 18 17.4 38.3 48.9 54.9 41.5 13.8 36.6 56.7 35.6 42.4 0 49.7 11 17.1 47.4 17.1 14.2 43.3 52.4 14.4 18.7 16.7 50.9 15.2 12.4 14.6 43.5 46.8 45.6 42 49.8 50.4 51.5 14.7 50.9 15 34.8 52.3 35.5 13.6 44.6 14.1 47.5 16.3 40.9 52.6 44 49.8 0 40.3 50.9 18.6 56.9 40.1 49.3 45 0 38.7 14.4 15.2 48.9 53.6 42.6 14.6 39 49.5 42.3 54.5 12.5 14.2 51.5 41 12.3 51.2 43.2 17.8 34.8 50.1 53 53.1 13.4 16.5 17.7 45 39.2 47.2 37.9 45.6 7.6 49.4 48.7 40.2 15.6 50.5 48.2 0 41.8 45.6 40.9 38.2 0 52.1 9.3 17.9 36.8 39.6 11.3 48.5 55.6 38.1 37.8 52.4 40.5 46.5 38.7 15.4 53.1 20.8 49.2 49.5 42.6 10.1 45.8 42.6 42.2 36.5 45.2 41.5 43.3 39.9 17.1 15.7 32.9 40.8 38.5 39.8 52.2 15.9 13.6 20.7 44 13.7 0 50 13.6 38.7 51.5 16.4 37.2 15.1.

He is interested in testing whether the mean height of sunflowers grown from KwikGrow seeds is greater than 33 mm two weeks after planting. He decides to conduct a hypothesis test by assuming that the sampling distribution of the sample mean has a normal distribution. For the purposes of this question, you may assume that the standard deviation of the sunflower heights is 13 mm.

a.Name one assumption that is required for the gardener to be able to use his sample to draw conclusions about KwikGrow seeds in general.

b.Name one key feature of the data that will assure the approximate validity of the gardener’s assumption that the sampling distribution of the sample mean has a normal distribution. Do you think his decision was wise considering the distribution of individual sunflower heights?

c.What are the null and alternative hypotheses?

d.What is the value of the test statistic?

e.What is the p-value (assuming the sample mean does indeed have a normal distribution)?

f.Using a significance level of alpha=0.05, state your conclusions in the language of the problem.

Answer 1

One-sample Kolmogorov-Smirnov test

D = 0.20244, p-value = 4.188e-11
alternative hypothesis: two-sided

So the observations do not follow normal distribution. However this is a continuous variable i.e.  seedlings’ heights and sample size is very large then from central limit theorem, we can approximate the sampling distribution of the sample mean by normal distribution.

and conclude that the mean height of sunflowers grown from KwikGrow seeds is greater than 33 mm two weeks after planting.