Hi guys/gals!
First time posting so apologies if I inadvertently cross any lines! I'm a Dutch substitute teacher in secondary/high school and I'm working on a problem that I don't have any answers for. I'll admit that the subject of probability and statistics isn't my strong suit and I'm working hard to improve. The question is as follows (translated, obviously):
The performance of competition speed skaters depends on their fitness, but also on external factors such as ice quality and weather. If a skater completes multiple races on the same track in one season, the effect of external factors can more or less be neglected. We therefore assume the results for 500m races to follow a normal distribution.
Benjamin is a young skater who always races on the same track in Utrecht. His 500m times in training follow a normal distribution with an average of 39.72 seconds and a standard deviation of 0.43 seconds.
1. Calculate the percentage of 500m times higher than 39 seconds.
Sabrina is another skater who also trains in Utrecht. Of the 100 races she completed over 500m, 25 of them were faster than 41 seconds. Her 500m times also follow a normal distribution with a standard deviation of 0.35 seconds.
2. Calculate her average 500m time. Round to two decimal places.
Many skaters find it advantageous to finish in the outer lane on the 500m. Speeds can be well over 50 km/h, making the final inner turn quite challenging because of the tighter bend. This is why skaters used to do two races of the 500m; once finishing on the inside and once on the outside. In a specific world championship 26 of the 40 skaters skated a faster time when finishing on the outside than they did when they finished on the inside.
3. Calculate whether this result is reason to assume skaters are generally faster on the 500m when they finish on the outside. Use a significance level of 5%.
My logic for these questions is as follows:
1. Define X as the result of a 500m race, following a normal distribution with average 39.72 and standard deviation of 0.43. We're looking for P(X<39.00) which I calculate on a casio fx-CG50 as follows:
P(X<39.00)=NormCD(-10^99, 39.00, 0.43, 39.72) which returns approximately 4.7%. This makes sense to me as 95% his 500m times should lie between 38.86 and 40.58.
2. Again, define X as the result of a 500m race, following a normal distribution with an unknown average and a standard deviation of 0.35 seconds. We have 100 samples, of which 25 were faster than 41 seconds. Using this, we calculate the sample standard deviation to be 0.35/sqrt(100)=0.035 seconds. We should look for an average that will result in that 25/100 result. So P(X<41.00)=NormCD(-10^99, 41.00, 0.035, µ)=0,25. Using the equation solver I get an approximate value for µ of 41.02 seconds. I'm not sure about this result tbh.
3. Here we assume a binomial distribution since skaters either skate faster when they finish outside or they do not. Because of this I also assume a p-value of 0.5. Define X as the number of skaters that performs better when finishing in the outer lane that follows a binomial distribution with n=40 samples. Because we're trying to see if the 26 skaters that were faster are a significant result, we should look for P(X>=26) (greater or equal to). This can be calculated by looking at 1-P(X<=25) and comparing that to our alpha. Performing the calculation we get P(X>=26)=1-BinomialCD(25, 40, 0.5) which is approximately 0.0403. This is less than alpha, meaning there's no reason to assume finishing in the outer lane makes a difference.
Are my results and explanations correct? Much obliged for any feedback!
First time posting so apologies if I inadvertently cross any lines! I'm a Dutch substitute teacher in secondary/high school and I'm working on a problem that I don't have any answers for. I'll admit that the subject of probability and statistics isn't my strong suit and I'm working hard to improve. The question is as follows (translated, obviously):
The performance of competition speed skaters depends on their fitness, but also on external factors such as ice quality and weather. If a skater completes multiple races on the same track in one season, the effect of external factors can more or less be neglected. We therefore assume the results for 500m races to follow a normal distribution.
Benjamin is a young skater who always races on the same track in Utrecht. His 500m times in training follow a normal distribution with an average of 39.72 seconds and a standard deviation of 0.43 seconds.
1. Calculate the percentage of 500m times higher than 39 seconds.
Sabrina is another skater who also trains in Utrecht. Of the 100 races she completed over 500m, 25 of them were faster than 41 seconds. Her 500m times also follow a normal distribution with a standard deviation of 0.35 seconds.
2. Calculate her average 500m time. Round to two decimal places.
Many skaters find it advantageous to finish in the outer lane on the 500m. Speeds can be well over 50 km/h, making the final inner turn quite challenging because of the tighter bend. This is why skaters used to do two races of the 500m; once finishing on the inside and once on the outside. In a specific world championship 26 of the 40 skaters skated a faster time when finishing on the outside than they did when they finished on the inside.
3. Calculate whether this result is reason to assume skaters are generally faster on the 500m when they finish on the outside. Use a significance level of 5%.
My logic for these questions is as follows:
1. Define X as the result of a 500m race, following a normal distribution with average 39.72 and standard deviation of 0.43. We're looking for P(X<39.00) which I calculate on a casio fx-CG50 as follows:
P(X<39.00)=NormCD(-10^99, 39.00, 0.43, 39.72) which returns approximately 4.7%. This makes sense to me as 95% his 500m times should lie between 38.86 and 40.58.
2. Again, define X as the result of a 500m race, following a normal distribution with an unknown average and a standard deviation of 0.35 seconds. We have 100 samples, of which 25 were faster than 41 seconds. Using this, we calculate the sample standard deviation to be 0.35/sqrt(100)=0.035 seconds. We should look for an average that will result in that 25/100 result. So P(X<41.00)=NormCD(-10^99, 41.00, 0.035, µ)=0,25. Using the equation solver I get an approximate value for µ of 41.02 seconds. I'm not sure about this result tbh.
3. Here we assume a binomial distribution since skaters either skate faster when they finish outside or they do not. Because of this I also assume a p-value of 0.5. Define X as the number of skaters that performs better when finishing in the outer lane that follows a binomial distribution with n=40 samples. Because we're trying to see if the 26 skaters that were faster are a significant result, we should look for P(X>=26) (greater or equal to). This can be calculated by looking at 1-P(X<=25) and comparing that to our alpha. Performing the calculation we get P(X>=26)=1-BinomialCD(25, 40, 0.5) which is approximately 0.0403. This is less than alpha, meaning there's no reason to assume finishing in the outer lane makes a difference.
Are my results and explanations correct? Much obliged for any feedback!