Data are drawn from a symmetric and bell-shaped distribution with a mean of 120 and a standard deviation of 5. There are 900 observations in the data set.

1. What percentage of observations are less than 130?
2. How many observations are less than 130?

The student is asking about the properties of a bell-shaped, symmetric distribution, particularly in regards to the percentage and the number of observations below a certain value. Given a mean of 120 and a standard deviation of 5, and the fact that the distribution is symmetric and bell-shaped, we can apply the Empirical Rule to determine the answers to the questions.
To find what percentage of observations are less than 130, we first need to determine how many standard deviations away from the mean this value is. Since the standard deviation is 5, 130 is two standard deviations above the mean (130 = 120 + 2*5). According to the Empirical Rule, approximately 95% of the data falls within two standard deviations of the mean in a normal distribution. Therefore, since our distribution is symmetric, about 97.5% of the data falls below 130 (because 50% is below the mean and 47.5% between one and two standard deviations above the mean).
To determine the actual number of observations less than 130, we take 97.5% of the total number of observations, which is 900. So, 0.975 * 900 = 877.5, and since we can't have a fraction of an observation, we would expect approximately 877 observations to be less than 130.

Answered by DestinyHopeCyrus | 2024-06-24

Approximately 97.5% of observations are less than 130, which translates to about 878 out of 900 observations in this symmetric, bell-shaped distribution.
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Answered by DestinyHopeCyrus | 2024-09-30