During a recent period of one year, the mean percent increase in value on Wednesdays of the cryptocurrency Dogecoin was 7.46%, with a standard deviation of 53.47%. Random samples of size 50 are drawn from this population and the mean of each sample is determined. (Source: Crypto Indicators)

c. What is the probability that the mean percent increase for a given sample is between −10% and 30%?

검증된 단계별 안내

Step 1: Identify the key parameters of the problem. The population mean (μ) is 7.46%, the population standard deviation (σ) is 53.47%, and the sample size (n) is 50. The goal is to find the probability that the sample mean (x̄) is between -10% and 30%.

Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is:

SE = \frac{σ}{\sqrt{n}}

. Substitute the values of σ = 53.47% and n = 50 into the formula.

Step 3: Standardize the bounds of the interval (-10% and 30%) to z-scores using the formula:

z = \frac{x̄ - μ}{SE}

. Compute the z-scores for both -10% and 30% by substituting the respective values of x̄, μ, and SE.

Step 4: Use the standard normal distribution table (or a statistical software) to find the cumulative probabilities corresponding to the z-scores calculated in Step 3. These cumulative probabilities represent the area under the standard normal curve up to each z-score.

Step 5: Subtract the cumulative probability of the lower z-score (corresponding to -10%) from the cumulative probability of the upper z-score (corresponding to 30%). This difference gives the probability that the sample mean is between -10% and 30%.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution. This is crucial for calculating probabilities related to sample means, especially when dealing with larger samples, such as the size of 50 in this question.

Standard Error

Standard Error (SE) measures the dispersion of sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the SE will help determine the probability of the sample mean falling within a specific range, such as between -10% and 30%.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the sample mean and dividing by the standard error. Z-scores are essential for finding probabilities in a normal distribution, allowing us to determine the likelihood of the sample mean being within the specified range.