A nationwide test taken by high school sophomores and juniors has three sections

Question

A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of 20 to 80. In a recent year, the national mean score for the writing section was 51.3 with a standard deviation of 9.2. Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year. (a) According to ChetysheVs theorem at least of the scores lie between 32.9 and 69.7 (b) According to Chebyshevs theorem, at least 36% of the scores lie between and (Round your answer to 1 decimal peace.) (c) Suppose that the distribution is bell-shaped According to the empirical rule approximately 68% of the sc exes lie between and (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately of the scores lie between 32.9 and 69.7.

Explanation / Answer

Excel formula:

a) limits 32.9 69.7 fraction of std deviation these values are away are= 2 2 so data should lie within 2+2=4 std deviation k=4 as per chebbyshev it the proportion data lies with this 4 *std dev limit is 1-1/k^2= 0.9375 b) Now prob is given i.e. 1-1/k^2=0.36 so k= 1.25 time std dev we for upper & lower limit we will use 1.25/2=0.625 std dev lower= 45.6 upper= 57.1 c) 68% of value is Normal distribution lies between +-1 SD lower= 42.1 upper= 60.5 D) limits 32.9 69.7 2 2 SO empirically about 95% fall in range of +-2 std dev exact figure= 0.954487

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