Final answer:
Using the empirical rule, we can determine the range of reading speeds for 68%, 95%, and 99.7% of second-grade students. The z-score calculation gives us a way to identify where a specific reading speed, such as 95 wpm, lies in relation to the average. Lastly, we estimate the probabilities for reading speeds to be more, less, or within specific ranges based on the standard normal distribution.
Step-by-step explanation:
The reading speed of second-grade students is approximately normal with a mean (μ) of 90 words per minute (wpm) and a standard deviation (σ) of 10 wpm.
a. Range for 68% of students
Based on the empirical rule:
90 ± 1(10) = [80, 100] wpm
b. Range for 95% of students
Based on the empirical rule:
90 ± 2(10) = [70, 110] wpm
c. Range for 99.7% of students
Based on the empirical rule:
90 ± 3(10) = [60, 120] wpm
d. Z-score for 95 wpm
Z = (95 - 90) / 10 = 0.5. This z-score means that reading 95 wpm is 0.5 standard deviations above the mean.
e. Probability of reading more than 95 wpm
This requires looking up the z-score in a standard normal distribution table, but approximately 30.85% read more than 95 wpm.
f. Probability of reading less than 80 wpm
Similarly, this would be 50% (below the mean) minus the percentage corresponding to 1 standard deviation above the mean, yielding approximately 15.85%.
g. Probability of reading less than 75 wpm or more than 100 wpm
This is outside the range of one standard deviation from the mean on both ends, so again using the standard normal table, the probability is around 32% (16% for each tail).
Learn more about Empirical Rules and Reading Speeds