A logging company is bidding for logging rights on a large tract ofland. Thegirthof a tree is the circumference of its trunk at a particular heightabove the ground (roughly breast height). The girth is important commercially, because it determines the width of the boards that can be milled from the tree. The owner of the property claims that the distribution of tree girth on theproperty is as follows: girth range<3 ft. 3-6 ft. 6-9 ft. 9-12ft >12 ft.
claimed fraction of trees10% 20% 35% 30% 5%
The logging company can estimate the number of trees fairly accurately usingaerial photography, but needs to do a survey on the ground to determine thedistribution of girth. Ground surveys are expensive, so the logging companywould like to draw its conclusions without surveying the entire tract.
The company hires you as an expert statistician to test the hypothesis that thedistribution of girth is what the seller claims. You select 20 locations on theproperty at random from a map, and send a surveyor to those 20 locations tomeasure all the trees within a 25 foot radius of the location.
Suppose that this results in measurementsof the girth of 340 trees, and that forall practical purposes, those 340 trees can be treated as a random sample withreplacement from the population of trees on the property. The observed distribution of girth among the 340 trees in the sample is asfollows:
girth range<3 ft. 3-6 ft. 6-9 ft. 9-12ft >12 ft.
observed number of trees 44 74 111 90 21
The observed value of the chi-squared statistic is:
Under the null hypothesis, the smallest expected number of trees inany of the categories isso, under the null hypothesis, theprobability histogram of the chi- well by a chi-squared curve. The appropriate chi-squared curve to use to approximate theprobability histogram of the chi-squared statistic has (enter a number)degrees of freedom. TheP-value of the null hypothesis is approximately.
Should the null hypothesis be rejected at significance level10%?