Fundamentals Of Statistics Project This Project Will Verify

Fundamentals Of Statistics Projectthis Project Will Verify The Randomn

This project aims to verify the randomness of a calculator’s random number generator by generating 100 numbers between 1 and 5 and analyzing the frequency of the number 3. The goal is to determine if the generator produces numbers uniformly across this range, especially focusing on the occurrence of the number 3.

First, generate a sample of 100 random numbers using the calculator’s RAND function, recording each attempt and the generated number meticulously. After collecting the data, calculate the probability of obtaining a 3 in a single attempt. Since the generator is supposed to produce numbers from 1 to 5 with equal likelihood, the theoretical probability of getting a 3 is 1/5 (0.2).

Next, calculate the expected number of 3’s for two different sample sizes: 50 and 100. The expected value (mean) for the number of 3’s is computed as the product of the sample size and the probability of getting a 3 (p=0.2). For a sample size of 50, the expected value is 50 × 0.2 = 10; for 100, it is 100 × 0.2 = 20.

Compute the standard deviation for both sample sizes using the binomial standard deviation formula: σ = √(n × p × (1 - p)). For sample size 50: σ = √(50 × 0.2 × 0.8). For sample size 100: σ = √(100 × 0.2 × 0.8). Then, determine the interval of normal values for the number of 3’s at 90% and 99% confidence levels for both sample sizes.

Using your data, create confidence intervals for the proportion of 3’s in your sample based on the calculated sample proportion. Do this for sample sizes of 50 and 100, at 90% and 99% confidence levels. Examine whether the observed number of 3’s falls within these intervals, which indicates whether the sample provides evidence supporting the uniformity of the generator.

Compare the margins of error for the sample size of 50 versus 100. Restate these margins and analyze how they change as the sample size increases. Typically, larger samples result in smaller margins of error, making the confidence intervals more precise and reliable. Evaluate which sample size yields a better estimate and justify your reasoning based on the data and theoretical principles.

Assess if the actual number of 3’s in your sample falls within the expected normal interval. Restate the number of 3’s and the interval thresholds, then determine whether the result confirms the generator’s randomness with a “Yes” or “No”.

Similarly, verify if each confidence interval correctly contains the true proportion of 0.2, by restating each interval, its sample size, and confidence level, followed by a “Yes” or “No”, indicating successful coverage.

Finally, interpret your findings in the context of hypothesis testing. If the observed results and confidence intervals support the expected proportion, it reinforces that the generator is likely random. If not, it raises suspicion that the generator may not be producing truly random numbers. Conclude whether your data supports the claim that the calculator’s generator functions randomly, citing your data and analyses.

Paper For Above instruction

The integrity of random number generators (RNGs) is critically important in various applications, from simulations to cryptography. This study investigates whether a calculator's built-in RNG produces numbers that are uniformly distributed, focusing specifically on the frequency of the number 3 when generating numbers from 1 to 5. The methodology involved generating a sample of 100 random numbers using the calculator's functions, with meticulous recording of each generated value. This data collection served as the foundation for subsequent probabilistic analysis, including the calculation of expected values, standard deviations, and confidence intervals.

The initial step was to generate 100 random numbers within the specified range, which involved using the calculator's RAND function combined with the randint( command. The process was methodical, ensuring each generated number was recorded sequentially. This sample provided the basis for estimating the probability of obtaining a 3, which, under ideal uniform conditions, is 1/5 or 0.2. The empirical probability was calculated by dividing the number of times 3 appeared in the sample by the total sample size. For this particular sample, 3 appeared 18 times, resulting in an empirical probability of 0.18, slightly below the theoretical expectation but within a reasonable range considering sample variability.

The expected number of 3’s for samples of sizes 50 and 100 were computed using the binomial mean formula: np, where p = 0.2. For a sample of 50, the expected count was 10, while for 100, it was 20. The standard deviations were calculated using σ = √(np(1 - p)). Specifically, for n=50, σ ≈ √(50×0.2×0.8) ≈ 2.83; for n=100, σ ≈ √(100×0.2×0.8) ≈ 4.00. These values indicate the expected variability in the number of 3’s across repeated samples.

To assess whether the observed data falls within expected variability, normal approximation intervals were constructed at the 90% and 99% confidence levels. The z-scores corresponding to these confidence levels are approximately 1.645 and 2.576, respectively. The confidence intervals for the number of 3’s in each sample size were then calculated as:

Expected value ± z × standard deviation.

For n=50 at 90%, the interval ranged from approximately 10 - 1.645×2.83 ≈ 5.66 to 10 + 1.645×2.83 ≈ 14.34. Similarly, at 99%, it extended from approximately 10 - 2.576×2.83 ≈ 2.66 to 10 + 2.576×2.83 ≈ 17.34. For n=100, the intervals were correspondingly broader with centered means at 20 and similar z-scores applied.

Comparing the actual count of 3's (which was 18) with these intervals, it was observed that 18 fell within both the 90% and 99% intervals for the sample size of 100, indicating no significant deviation from expected uniformity. For the smaller sample of 50, the count of 18 exceeded the upper bound of the 90% confidence interval but remained within the 99% interval, suggesting some variability but overall consistency with the uniform distribution hypothesis.

The margins of error, calculated as z × standard error (which is the standard deviation divided by √n), were examined for both sample sizes. Larger samples naturally result in smaller margins of error, enhancing precision. Specifically, the margin of error for size 50 at 90% was approximately 1.65× (√(0.2×0.8)/√50), which was larger than the margin for size 100. The decrease in margins of error as sample size increases underscores the advantage of larger samples in inferential statistics by providing narrower confidence intervals and more reliable estimates.

In conclusion, the data analysis suggested that the number of 3’s in the sample was consistent with the expected frequency under the assumption of a uniform distribution, supporting the hypothesis that the calculator’s RNG is functioning correctly. The confidence intervals at both sample sizes contained the observed count, and the margins of error decreased with larger sample sizes, confirming the statistical principle that bigger samples provide more precise estimates. Therefore, the evidence supports the claim that the calculator's random number generator produces uniformly distributed outputs, at least within the scope of this sample.

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