On a test of mathematical ability, Jia scored 20 out of 40. Her friend Sayesha scored 25. We know that the class scores are normally distributed with a mean of 28 and a standard deviation of 3. Jia and Sayesha’s friend Swastik is away when they get their marks. He is worried about doing poorly. What is the probability of Swastik obtaining a score between that of Jia and Sayesha?
Question
On a test of mathematical ability, Jia scored 20 out of 40. Her friend Sayesha scored 25. We know that the class scores are normally distributed with a mean of 28 and a standard deviation of 3. Jia and Sayesha’s friend Swastik is away when they get their marks. He is worried about doing poorly. What is the probability of Swastik obtaining a score between that of Jia and Sayesha?
Solution
To solve this problem, we need to convert the scores of Jia and Sayesha into z-scores, which are a measure of how many standard deviations an element is from the mean.
The formula for calculating a z-score is:
Z = (X - μ) / σ
where: X is the raw score, μ is the population mean, and σ is the standard deviation.
First, let's calculate the z-scores for Jia and Sayesha:
For Jia: Z_Jia = (20 - 28) / 3 = -8 / 3 = -2.67
For Sayesha: Z_Sayesha = (25 - 28) / 3 = -3 / 3 = -1
Now, we need to find the probability that Swastik's score falls between these two z-scores.
We can look up these z-scores in a standard normal distribution table, or use a calculator or software that can calculate it.
The probability for Z = -1 is approximately 0.1587 (or 15.87%) and for Z = -2.67 is approximately 0.0038 (or 0.38%).
The probability of Swastik scoring between Jia and Sayesha is the difference between these two probabilities:
P = 0.1587 - 0.0038 = 0.1549 or 15.49%
So, there is approximately a 15.49% chance that Swastik will score between Jia and Sayesha.
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