Question

2. Some scientists will measure the speed of light many times with a method resulting in measurements having an expected value of 299,774 kilometers per second (km/s) and standard deviation 14 km/s. The true speed of light is 299,792 km/s (you will notice I have rounded if you fact-check this) (a) Assume the measurements are Normally distributed i. What proportion of their measurements will exceed 299,792 km/s? ii. Give the value such that 95% of their measurements will lie above it. iii. Give two values such that 80% of the measurements will lie between them. iv. If they take 50 measurements, give the probability that 10 or more exceed 299,792 km/s. (b) Assume the measurements are not Normally distributed. The scientists take 50 measurements of the speed of light and record the average as X i. With approximately what probability will X, exceed the true speed of light? ii. With approximately what probability does Xlie within 20 km/s of the true speed of light? iii. Give an interval within which X, will lie with probability approximately 95%. iv. Give the value such that X, exceeds it with probability approximately 0.01. v. What theorem did you invoke to justify your calculations?

Answer 1

For normally distributed measurements, we calculated probabilities and intervals based on Z-scores and binomial distribution. For non-normally distributed measurements, we provided general insights, emphasizing the central limit theorem's role in the normal distribution approximation.

(a) Measurements are Normally distributed

i) Proportion exceeding 299,792 km/s:

Z-score for 299,792 km/s: (299,792 - 299,774) / 14 = 0.136

Cumulative area to the right of 0.136 in a standard normal distribution: ~0.5517

Proportion exceeding 299,792 km/s: 1 - 0.5517 = 0.4483 (44.83%)

ii) 95% of measurements above a value:

Z-score for 95% cumulative area: ~1.645 (corresponding to 2.5% in each tail)

Value above which 95% measurements lie: 299,774 + (1.645 * 14) = 299,820.2 km/s

iii) 80% of measurements between two values:

We need to find two Z-scores such that the area between them covers 80% of the standard normal distribution. This requires finding the Z-scores corresponding to 10% in each tail (since 80% + 10% + 10% = 100%).

These Z-scores are approximately -1.282 and 1.282.

Corresponding values in km/s: 299,774 - (1.282 * 14) = 299,738.2 km/s and 299,774 + (1.282 * 14) = 299,809.8 km/s

iv) Probability of 10 or more exceeding 299,792 km/s in 50 measurements:

This is a binomial probability calculation.

Probability of exceeding 299,792 km/s with one measurement: 0.4483 (from part i)

We want the probability of 10 or more exceeding it in 50 trials.

Using the binomial probability formula: P(X >= 10) = 1 - P(X < 10) = 1 - binomcdf(50, 0.4483, 9) ≈ 0.9932

Therefore, the probability of 10 or more measurements exceeding the true speed of light is 0.9932 (99.32%).

(b) Measurements are not Normally distributed

i) Probability of X exceeding the true speed of light:

Without knowing the specific distribution, we cannot calculate this probability precisely. However, we can say that it is less than 0.5. This is because the average of any distribution tends towards the center of its mass, and the true speed of light is above the average of the known distribution (299,774 km/s).

ii) Probability of X within 20 km/s of the true speed of light:

Similarly, without knowing the specific distribution, we cannot calculate this probability precisely. However, we can say that it is greater than 0. This is because any distribution will have some spread, and the true speed of light lies within 20 km/s of the average in this case.

iii) Interval for X with 95% probability:

We cannot determine the exact interval without knowing the distribution. However, we can say that it will be centered around 299,774 km/s and have a width smaller than 40 km/s (twice the standard deviation).

iv) Value X exceeds with 0.01 probability:

Again, we cannot determine the exact value without knowing the distribution. However, it will be higher than 299,774 km/s and likely more than 40 km/s above the average.

v) Theorem:

We used the central limit theorem for part (a), assuming the measurements are normally distributed. This theorem states that the sum of a large number of independent random variables, regardless of their individual distributions, tends towards a normal distribution as the number of variables increases.

This allows us to approximate the distribution of the average with a normal distribution even if the individual measurements are not normally distributed.