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Answer :
Final answer:
To find the proportion of student heights between 93 cm and 100.5 cm, calculate the z-scores of both values using the given mean and standard deviation, and then use a standard normal distribution table to find the corresponding proportions.
Explanation:
To determine the proportion of elementary school student heights that are between 93 centimeters and 100.5 centimeters given that the heights are normally distributed with a mean of 105 centimeters and a standard deviation of 5 centimeters, we need to calculate the z-scores for both 93 centimeters and 100.5 centimeters and then find the corresponding proportions using the standard normal distribution (z-distribution).
The z-score for a value X is given by the formula:
Z = (X - mean) / standard deviation
First, we calculate the z-score for 93 centimeters:
Z = (93 - 105) / 5 = -12 / 5 = -2.4
Then, we calculate the z-score for 100.5 centimeters:
Z = (100.5 - 105) / 5 = -4.5 / 5 = -0.9
Using a standard normal distribution table or calculator, we can find the proportion of the population below each z-score and then subtract the smaller proportion from the larger proportion to find the percentage of student heights that fall between these two scores.
Without the actual table values, we can't provide a numerical answer, but this method will give you the needed proportion.
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Final answer:
Calculate the Z-scores for 93 and 100.5 centimeters, then use a standard normal distribution table or a calculator to find the proportion of student heights between these two values.
Explanation:
To determine the proportion of elementary school student heights that are between 93 centimeters and 100.5 centimeters given that the heights are normally distributed with a mean of 105 centimeters and a standard deviation of 5 centimeters, we need to calculate the z-scores for both 93 centimeters and 100.5 centimeters and then find the corresponding proportions using the standard normal distribution (z-distribution).
The z-score for a value X is given by the formula:
Z = (X - mean) / standard deviation
First, we calculate the z-score for 93 centimeters:
Z = (93 - 105) / 5 = -12 / 5 = -2.4
Then, we calculate the z-score for 100.5 centimeters:
Z = (100.5 - 105) / 5 = -4.5 / 5 = -0.9
Using a standard normal distribution table or calculator, we can find the proportion of the population below each z-score and then subtract the smaller proportion from the larger proportion to find the percentage of student heights that fall between these two scores.
Without the actual table values, we can't provide a numerical answer, but this method will give you the needed proportion.
