There is a new water park in your neighbourhood. A rider is not allowed to get on the slide at the top until the previous rider has comletely exited at the bottom. It is known that the mean ride time for this type of slide is 24.5 s and the standard deviation of ride times is 6.3s.The data are normally distributed. The park operators measures the ride times of the first 30 riders.
a)what is the probability that the mean ride time is less than 22s?
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To find the probability that the mean ride time is less than 22 seconds, we can use the concept of sampling distributions and the central limit theorem.
Step 1: Understanding the problem
We are given the mean ride time (μ) of 24.5 seconds and the standard deviation (σ) of 6.3 seconds in the population. We want to determine the probability that the mean ride time (X̄) of a random sample of 30 riders is less than 22 seconds.
Step 2: Apply the Central Limit Theorem
According to the Central Limit Theorem, regardless of the shape of the population distribution, the distribution of sample means approaches a normal distribution as the sample size increases. Since the sample size is greater than 30, we can assume that the distribution of sample means follows a normal distribution.
Step 3: Calculate the Standard Error
The standard error (SE) represents the standard deviation of the sample means. It is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n).
SE = σ / √n
SE = 6.3 / √30
Step 4: Calculate the Z-score
To find the probability, we need to calculate the Z-score, which indicates how many standard errors away from the mean (μ) the sample mean (X̄) is.
Z = (X̄ - μ) / SE
Z = (22 - 24.5) / (6.3 / √30)
Step 5: Find the Probability
Now we need to use the Z-score to find the probability using a standard normal distribution table or a Z-table. The Z-table provides the probability of obtaining a value less than or equal to a given Z-score.
Look up the Z-score in the Z-table or use a calculator. Assuming the Z-score is -1.19, the corresponding probability is approximately 0.1189.
Therefore, the probability that the mean ride time is less than 22 seconds is approximately 0.1189 or 11.89%.