What is the probability that a random sample of 25 swimmers results in a sample mean of 49 seconds or less?
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To determine the probability that a random sample of 25 swimmers results in a sample mean of 49 seconds or less, we need to know the distribution of swim times and its parameters. Without this information, we cannot directly calculate the probability.
However, if we assume that the swim times are normally distributed with a known mean (μ) and standard deviation (σ), we can use the Central Limit Theorem to calculate an approximate probability.
The Central Limit Theorem states that if we have a large enough sample size, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution. For sample sizes greater than or equal to 30, the approximation is usually reliable.
Here is a step-by-step approach to estimating the probability using the Central Limit Theorem:
1. Determine the mean and standard deviation of the swim times (population parameters). If these are not provided, you will need to obtain them from the available data or assumptions.
2. Calculate the standard error (SE) of the sample mean using the formula: SE = σ / √n, where σ is the population standard deviation and n is the sample size. In this case, n = 25.
3. Standardize the sample mean by subtracting the population mean (μ) and dividing by the standard error: Z = (x - μ) / SE, where x is the sample mean.
4. Look up the corresponding probability in the standard normal distribution table (also known as the Z-table) for the standardized value (Z). This gives you the probability that the sample mean is less than or equal to a certain value (in this case, 49 seconds).
Keep in mind that this approach assumes certain conditions such as independence of observations and random sampling. If these conditions are not met, adjustments may be necessary for a valid estimation.
Without specific values for the mean, standard deviation, and the assumption of the normal distribution, we cannot provide a specific probability in this case.