The Edmonton rush is one of nine teams that play in the national lacrosse league. The weights of the 2013 Edmonton rush players are shown.
Weight in Lbs.
160. 170. 170. 175. 180
185. 188. 190. 190. 190
194. 195. 200. 200. 200
205. 205. 210. 210. 210
210. 215. 245.
A.) Explain why this data is close to being normally distributed.
B.) There were 218 players in the national lacrosse league in 2013. Assuming the mean and standard deviation of the league are same as the mean and standard deviation of the Edmonton rush, predict the number of players that weighed over 231 lbs.
C.) Below what weight should there be approximately 35 national lacrosse league players?
Please help, I have no idea how to do these.
B) Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 218.
C) Find Z score probability = 35/218. Insert Z value into above equation.
A.) To determine if the data is close to being normally distributed, we can analyze the shape and characteristics of the data. The Edmonton Rush's weight data appears to be close to a normal distribution because:
1. There is a central peak in the distribution, with the majority of weights falling around the mean.
2. The data shows a symmetric pattern, meaning that the weights are distributed evenly around the mean.
3. There are no extreme outliers or skewed values that significantly affect the distribution.
4. The range of weights does not vary greatly, as there are no sudden drops or spikes in the data.
These characteristics suggest that the data is close to being normally distributed.
B.) To predict the number of players weighing over 231 lbs, we need to calculate the z-score and use it to find the corresponding percentile. With the given data, we can calculate the mean and standard deviation of the Edmonton Rush's weight distribution. The mean can be calculated by summing all the weights and dividing by the total number of players (23 in this case). The standard deviation can be calculated using the following formula:
Standard deviation = √ [Σ(x - μ)² / N]
where Σ represents the sum, x is each weight, μ is the mean, and N is the total number of weights.
Once we have the mean and standard deviation, we can calculate the z-score for a weight of 231 lbs using the formula:
z-score = (x - μ) / σ
where x is the weight we want to find the z-score for, μ is the mean, and σ is the standard deviation.
Finally, we can use a z-score table to find the percentile corresponding to the z-score. This percentile represents the proportion of players weighing over 231 lbs.
C.) To determine the weight below which there would be approximately 35 national lacrosse league players, we can use a z-score table and the given mean and standard deviation of the league.
First, we need to find the z-score that corresponds to the percentile associated with approximately 35 players. To find this percentile, we subtract 35 player count from the total number of players (218), divide it by the total number of players, and multiply by 100 to obtain the percentile.
Next, we can use the z-score table to find the z-score associated with the obtained percentile.
Finally, we can use the z-score formula to solve for the weight below which there are approximately 35 players:
x = μ + (z * σ)
where x is the weight we want to determine, μ is the mean, z is the z-score, and σ is the standard deviation.
A.) To explain why this data is close to being normally distributed, we need to consider the characteristics of a normal distribution. A normal distribution, also known as a bell curve, is symmetrical and follows a specific pattern where most of the data points cluster around the mean, with fewer data points farther away from the mean.
In this case, the weights of the 2013 Edmonton Rush players are listed. By examining the data, we can see that the weights are distributed relatively evenly around the middle values, which are around 190lbs. The weights gradually increase from around 160lbs to 245lbs without any significant gaps or sudden jumps in values. Additionally, the distribution appears to be relatively symmetrical, as there are similar numbers of players on both sides of the mean.
B.) To predict the number of players that weighed over 231lbs, we need to use the concept of the standard normal distribution and the Z-score. The Z-score is a measure of how many standard deviations a data point is away from the mean.
First, we need to calculate the Z-score for 231lbs using the mean and standard deviation of the Edmonton Rush data. To do this, we use the formula:
Z-score = (x - mean) / standard deviation
Assuming we have the mean and standard deviation values, we can calculate the Z-score for 231lbs. Let's assume the mean is 195lbs and the standard deviation is 15lbs.
Z-score = (231 - 195) / 15
Z-score = 2.4
Once we have the Z-score, we can refer to a standard normal distribution table or use statistical software to determine the proportion of data points above 231lbs. The Z-score of 2.4 corresponds to approximately 0.9918 on the standard normal distribution. This means that about 99.18% of the data is below 231lbs.
To predict the number of players weighing over 231lbs, we subtract the proportion below 231lbs from 1 (the total proportion) and multiply it by the total number of players in the league.
Number of players over 231lbs = (1 - 0.9918) * 218
Number of players over 231lbs ≈ 1.82
Based on this calculation, we can predict that there would be approximately 1 or 2 players weighing over 231lbs in the league.
C.) To determine the weight below which there should be approximately 35 national lacrosse league players, we need to utilize the cumulative distribution function (CDF) of the normal distribution.
Using the same mean and standard deviation from the Edmonton Rush data, we need to find the weight value that corresponds to a cumulative probability of 35%. This means we want to find the weight below which approximately 35% of the players fall.
To find this weight, we can refer to a standard normal distribution table or use statistical software to identify the Z-score that corresponds to a cumulative probability of 35%. Let's assume that the Z-score corresponds to -0.385.
Using the formula for the Z-score:
Z-score = (x - mean) / standard deviation
Rearranging the formula, we can solve for x:
x = mean + (Z-score * standard deviation)
Assuming the mean is 195lbs and the standard deviation is 15lbs, we can calculate the weight.
x = 195 + (-0.385 * 15)
x = 188.225
Therefore, there should be approximately 35 national lacrosse league players with a weight below 188.225lbs.