In: Statistics and Probability
A medical researcher is interested in whether calcium intake differs, on average, among elderly individuals with normal bone density, with osteopenia (a low bone density that may lead to osteoporosis), and with osteoporosis. He recruits consenting 60 individuals over the age of 65 using his patient records from his hospital (20 from each of the three bone density groups). You may assume that the individuals in each group constitute a simple random sample from consenting patients from that hospital. The individuals keep a food diary for two weeks, at the end of which the researcher computes their average daily calcium intake (in mg). The data are available in the file calcium.csv.
CALCIUM.CSV:
calcium group 1 861 normal 2 884 normal 3 1009 normal 4 905 normal 5 909 normal 6 1020 normal 7 932 normal 8 811 normal 9 852 normal 10 869 normal 11 986 normal 12 925 normal 13 928 normal 14 908 normal 15 861 normal 16 1025 normal 17 935 normal 18 762 normal 19 949 normal 20 867 normal 21 675 osteopenia 22 735 osteopenia 23 678 osteopenia 24 699 osteopenia 25 706 osteopenia 26 632 osteopenia 27 809 osteopenia 28 761 osteopenia 29 670 osteopenia 30 838 osteopenia 31 780 osteopenia 32 729 osteopenia 33 813 osteopenia 34 811 osteopenia 35 808 osteopenia 36 798 osteopenia 37 789 osteopenia 38 746 osteopenia 39 729 osteopenia 40 723 osteopenia 41 651 osteoporosis 42 685 osteoporosis 43 611 osteoporosis 44 852 osteoporosis 45 785 osteoporosis 46 621 osteoporosis 47 672 osteoporosis 48 667 osteoporosis 49 755 osteoporosis 50 694 osteoporosis 51 718 osteoporosis 52 698 osteoporosis 53 697 osteoporosis 54 796 osteoporosis 55 684 osteoporosis 56 806 osteoporosis 57 592 osteoporosis 58 741 osteoporosis 59 709 osteoporosis 60 715 osteoporosis
a. What are the null and alternative hypotheses?
b. (1 mark) What is the value of the test statistic?
c. (1 mark) What is the p-value?
d. Using a significance level of α = 0.05, state your conclusions in the language of the problem.
e. State and verify (using plots and/or descriptive statistics) the additional two assumptions required for the p-value in c) to be valid.
here we use F-test using one-way anova
(a) Null hypothesis H0:normal=osteopenia=osteoporosis
alternate hypothesis Ha: atleast one is different from others
(b) F=55.211 ( the F-value of between group )
(c) p-value=0.000 (4.61E-14)
(d) Since p-value is less than typical alpha=0.05, so we reject the null hypothesis and conclude that Yes, calcium intake differs, on average, among elderly individuals with normal bone density, with osteopenia (a low bone density that may lead to osteoporosis), and with osteoporosis.
Anova: Single Factor | ||||||
SUMMARY | ||||||
Groups | Count | Sum | Average | Variance | ||
Column 1 | 20 | 18198 | 909.9 | 4641.463 | ||
Column 2 | 20 | 14929 | 746.45 | 3389.208 | ||
Column 3 | 20 | 14149 | 707.45 | 4507.208 | ||
ANOVA | ||||||
Source of Variation | SS | df | MS | F | P-value | F crit |
Between Groups | 461486 | 2 | 230743 | 55.21102 | 4.61E-14 | 3.158843 |
Within Groups | 238219.7 | 57 | 4179.293 | |||
Total | 699705.7 | 59 |