Question

The following data were collected on 300 8-year old girls. This data was then organized into categories (also called intervals) of interest. Test, at a .05 level of significance, that the data are drawn from a normally distributed population. The sample mean and standard deviation computed from the grouped data were 127.02 and 5.08, respectively. Observed Frequency Observed Frequency 43 Height in Centimerters 114 to 115.9 116 to 117.9 118 to 119.9 120 to 121.9 122 to 123.9 124 to 125.9 126 to 127.9 Height in Centimerters 128 to 129.9 130 to 131.9 132 to 133.9 134 to 135.9 136 to 137.9 138 to 139.9 Total 300

Answer 1

ANSWER:

First, we will calculate the probability & expected frequencies for each of the ranges given in the table. This can be calculated from the z-table and the expected frequency (normal distribution) can be calculated by multiplying the probability with 300 P(X1 <X<X2) Ei (Probability 300) X2 X1 115.9 0.009111 114 2.73 116 117.9 0.021275 6.38 118 119.9 0.042621 12.79 120 121.9 0.073255 21.98 122 123.9 0.10802 32.41 124 125.9 0.136659 41.00 126 127.9 0.148332 44.50 128 129.9 0.138132 41.44 130 131.9 0.110362 33.11 132 133.9 0.07565 22.69 134 135.9 0.044489 13.35 136 137.9 0.022447 6.73 138 139.9 0.009716 2.91

Ho: The data is drawn from a normal distribution Ha: The data is not drawn from a normal distribution Now, let's calculate the chi-square statistic (0-E,) Ei The calculations are shown below

(Oi-Ei) (Oi-Ei) (Oi-Ei) (Oi-EiyEi X1 X2 P(X1 <X<X2) Ei (Probability 300) Oi (Oi-Ei) 114 115.9 0.0091 2.73 2.27 5.139 5 1.880 116 117.9 0.0213 6.38 10 3.62 13.087 2.051 119.9 0.0426 1.473 118 12.79 14 1.21 0.115 120 121.9 0.0733 21.98 21 0.98 0.953 0.043 122 123.9 0.1080 -2.41 0.179 32.41 5.789 124 125.9 0.1367 41.00 0.024 -1.00 0.995 126 127.9 0.1483 44.50 45 0.50 0.251 0.006 128 129.9 0.1381 1.56 41.44 43 2.435 0.059 2.388 130 131.9 0.1104 33.11 42 8.89 79.055 132 133.9 0.0756 7.31 22.69 30 53.364 2.351 134 135.9 0.0445 5.507 0.413 13.35 11 -2.35 3.007 136 137.9 0.0224 6.73 5 -1.73 0.446 138 139.9 0.0097 2.91 4. 1.09 1.178 0.404 Total 10.359 Chi-square statistic 10.359 30 40

df 13-1 12 At df 12 and alpha 0.05, critical value 21.03 As the chi-square statistic 10.359 <21.03, hence we cannot reject the null hypothesis. We can conclude that the data is drawn from a normal distribution at 0.05 significance level.