Central Limit Theorem Probability Question with Pulse Rates in StatCrunch | Summary and Q&A

TL;DR

The content provides step-by-step instructions on calculating the probability of pulse rates for adult females using a normal distribution.

Key Insights

• ☠️ The mean pulse rate for adult females is 74 beats per minute, with a standard deviation of 12.5 minutes.
• ☠️ The probability of a randomly selected adult female having a pulse rate less than 80 beats per minute is 0.6844.
• ☠️ When dealing with the mean pulse rates of 25 randomly selected adult females, the new standard deviation is 2.5, and the probability of the mean pulse rates being less than 80 beats per minute is 0.9918.

Transcript

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Questions & Answers

Q: How do you calculate the probability of an adult female's pulse rate being less than 80 beats per minute?

To calculate this probability, we use the normal distribution formula with a mean of 74 and a standard deviation of 12.5. By inputting these values into a statistical calculator like StatCrunch, we find the probability to be 0.6844.

Q: How do you calculate the probability of the mean pulse rates for 25 randomly selected adult females being less than 80 beats per minute?

In this case, since we are dealing with the average (X bar), we need to compute the new standard deviation first. Using the formula Sigma/square root of n, where Sigma is 12.5 and n is 25, we find the new standard deviation to be 2.5. With this new value, we can input it into StatCrunch, along with the mean of 74 and the value of 80, to find the probability of 0.9918.

Q: Why can the normal distribution be used in the second scenario, even though the sample size is below 30?

In this scenario, the population was specified to be normal. According to statistical principles, if the original population has a normal distribution, the distribution of sample means will also be normal. Therefore, the normal distribution can be used in this case.

Q: What happens if the sample size exceeds 30?

When the sample size exceeds 30, the central limit theorem applies. This theorem states that if the sample size is larger than 30, the sample mean (X bar) will be approximately normally distributed. Thus, the normal distribution can be used in such cases.

Summary & Key Takeaways

• The content explains how to calculate the probability of an adult female's pulse rate being less than 80 beats per minute using a normal distribution.

• It also demonstrates how to calculate the probability of the mean pulse rates for 25 randomly selected adult females being less than 80 beats per minute.

• The content emphasizes the importance of computing the new standard deviation when dealing with sample averages.

• It addresses the question of why the normal distribution can be used for the second scenario, even though the sample size is below 30.