Studdy AI Logo
Math

QuestionIn 37 trials of the 400m dash, how many times will Daniel finish between 60 and 63 seconds, given μ=63\mu = 63 and σ=1.5\sigma = 1.5?

Studdy Solution

STEP 1

Assumptions1. Daniel's finishing times for the400 meter dash are normally distributed. . The mean of his finishing times is63 seconds.
3. The standard deviation of his finishing times is1.5 seconds.
4. Daniel runs37 practice trials.
5. We want to find out how many of these trials would have a finishing time between60 and63 seconds.

STEP 2

First, we need to standardize the values of60 and63 seconds to z-scores. The z-score is calculated by subtracting the mean from the value and then dividing by the standard deviation.
Z=Value−MeanStandard DeviationZ = \frac{Value - Mean}{Standard\, Deviation}

STEP 3

Now, plug in the given values for the mean, standard deviation, and the lower value (60 seconds) to calculate the lower z-score.
Zlower=60−631.5Z_{lower} = \frac{60 -63}{1.5}

STEP 4

Calculate the lower z-score.
Zlower=60−631.=−2Z_{lower} = \frac{60 -63}{1.} = -2

STEP 5

Now, plug in the given values for the mean, standard deviation, and the upper value (63 seconds) to calculate the upper z-score.
Zupper=63−631.5Z_{upper} = \frac{63 -63}{1.5}

STEP 6

Calculate the upper z-score.
Zupper=63−631.5=0Z_{upper} = \frac{63 -63}{1.5} =0

STEP 7

Now that we have the z-scores, we can find the proportion of trials that fall between these scores. We can use a standard normal distribution table or a calculator with a normal distribution function to find these proportions.
The proportion for the lower z-score is(Z < -2) and for the upper z-score is(Z <0).

STEP 8

The proportion of trials for the upper z-score ((Z <0)) is0.5 (as it is the mean of the standard normal distribution).

STEP 9

The proportion of trials for the lower z-score ((Z < -2)) can be found from the standard normal distribution table or calculator. It is approximately.0228.

STEP 10

Now, subtract the proportion of the lower z-score from the upper z-score to find the proportion of trials that fall between60 and63 seconds.
Proportion=(Z<0)−(Z<−2)Proportion =(Z <0) -(Z < -2)

STEP 11

Plug in the values for the proportions to calculate the proportion of trials between60 and63 seconds.
Proportion=0.5−0.0228Proportion =0.5 -0.0228

STEP 12

Calculate the proportion of trials between60 and63 seconds.
Proportion=0.5−0.0228=0.4772Proportion =0.5 -0.0228 =0.4772

STEP 13

Now that we have the proportion of trials that fall between60 and63 seconds, we can multiply this by the total number of trials (37) to find out how many of these trials would have a finishing time between60 and63 seconds.
Number of trials=ProportiontimesTotal number of trialsNumber\, of\, trials = Proportion \\times Total\, number\, of\, trials

STEP 14

Plug in the values for the proportion and the total number of trials to calculate the number of trials between60 and63 seconds.
Number of trials=0.4772times37Number\, of\, trials =0.4772 \\times37

STEP 15

Calculate the number of trials between60 and63 seconds and round to the nearest whole number.
Number of trials=0.4772times37≈18Number\, of\, trials =0.4772 \\times37 \approx18Daniel would have approximately18 trials with a finishing time between60 and63 seconds.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord