A distribution of values is normal with a mean of 110 and a standard deviation of 5.
Find the interval containing the middle-most 48% of scores:
Enter your answer using interval notation. Example: [2.1,5.6)
Your numbers should be accurate to 1 decimal places.
Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
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What does deviation mean?
To find the interval containing the middle-most 48% of scores in a normal distribution, we need to find the z-scores that correspond to the lower and upper tails of 24% each.
1. Start by finding the z-score corresponding to the lower tail:
Since the lower tail is 24%, we need to find the z-score that corresponds to a cumulative probability of 0.24. Using a standard normal distribution table or a calculator, you can find that the z-score for a cumulative probability of 0.24 is approximately -0.71.
2. Next, find the z-score corresponding to the upper tail:
Since the upper tail is also 24%, we need to find the z-score that corresponds to a cumulative probability of 0.76 (1 - 0.24). Using the same standard normal distribution table or calculator, you can find that the z-score for a cumulative probability of 0.76 is approximately 0.75.
3. Now, we can convert the z-scores to raw scores using the formula:
raw score = z-score * standard deviation + mean
Lower bound: -0.71 * 5 + 110 = 106.45 (rounded to 1 decimal place)
Upper bound: 0.75 * 5 + 110 = 113.75 (rounded to 1 decimal place)
Therefore, the interval containing the middle-most 48% of scores is [106.5, 113.8).