How To Perform The Following Using TI-84 Plus Calculator: A New Surgery Is Successful 85% Of The Time. If The Results Of 6 Such Surgeries Are Randomly Sampled, What Is The Probability That At Least 4 Of Them Are Successful? Carry Your Intermediate

by ADMIN 248 views

Introduction

The TI-84 Plus calculator is a powerful tool for performing various mathematical calculations, including hypothesis testing and probability calculations. In this article, we will guide you through the steps to perform a hypothesis test and calculate the probability of at least 4 successful surgeries out of 6, given that a new surgery is successful 85% of the time.

Understanding the Problem

A new surgery is successful 85% of the time. If the results of 6 such surgeries are randomly sampled, what is the probability that at least 4 of them are successful? To solve this problem, we will use the binomial probability formula, which is given by:

P(X = k) = (nCk) * (p^k) * (q^(n-k))

where:

  • P(X = k) is the probability of k successes
  • n is the number of trials (6 surgeries in this case)
  • k is the number of successes (at least 4)
  • p is the probability of success (0.85)
  • q is the probability of failure (1 - p = 0.15)
  • nCk is the number of combinations of n items taken k at a time

Using the TI-84 Plus Calculator

To perform the calculation on the TI-84 Plus calculator, follow these steps:

Step 1: Enter the values

  1. Press the STAT button to access the statistics menu.
  2. Select 1: 3:Binomcdf to access the binomial cumulative distribution function.
  3. Enter the values:
    • n (number of trials): 6
    • p (probability of success): 0.85
    • k (number of successes): 4 (we will calculate the probability for at least 4 successes)
    • lower (lower bound): 0 (we want to calculate the probability from 0 to 4 successes)
    • upper (upper bound): 6 (we want to calculate the probability up to 6 successes)

Step 2: Calculate the probability

  1. Press the ENTER button to calculate the probability.
  2. The calculator will display the probability of at most 4 successes.

Step 3: Calculate the probability of at least 4 successes

  1. To calculate the probability of at least 4 successes, we need to subtract the probability of at most 3 successes from 1.
  2. Press the STAT button to access the statistics menu.
  3. Select 1: 3:Binomcdf to access the binomial cumulative distribution function.
  4. Enter the values:
    • n (number of trials): 6
    • p (probability of success): 0.85
    • k (number of successes): 3
    • lower (lower bound): 0
    • upper (upper bound): 3
  5. Press the ENTER button to calculate the probability.
  6. The calculator will display the probability of at most 3 successes.
  7. Subtract this value from 1 to get the probability of at least 4 successes.

Calculating the Probability

Using the TI-84 Plus calculator, we get:

P(X ≤ 3) = 0.9999

To calculate the probability of at least 4 successes, we subtract this value from 1:

P(X ≥ 4) = 1 - P(X ≤ 3) = 1 - 0.9999 = 0.0001

Therefore, the probability that at least 4 of the 6 surgeries are successful is approximately 0.0001 or 0.01%.

Conclusion

In this article, we demonstrated how to perform a hypothesis test and calculate the probability of at least 4 successful surgeries out of 6, given that a new surgery is successful 85% of the time. We used the binomial probability formula and the TI-84 Plus calculator to perform the calculation. The probability of at least 4 successes is approximately 0.0001 or 0.01%.

Q: What is the binomial probability formula?

A: The binomial probability formula is given by:

P(X = k) = (nCk) * (p^k) * (q^(n-k))

where:

  • P(X = k) is the probability of k successes
  • n is the number of trials
  • k is the number of successes
  • p is the probability of success
  • q is the probability of failure
  • nCk is the number of combinations of n items taken k at a time

Q: How do I enter the values on the TI-84 Plus calculator?

A: To enter the values on the TI-84 Plus calculator, follow these steps:

  1. Press the STAT button to access the statistics menu.
  2. Select 1: 3:Binomcdf to access the binomial cumulative distribution function.
  3. Enter the values:
    • n (number of trials)
    • p (probability of success)
    • k (number of successes)
    • lower (lower bound)
    • upper (upper bound)

Q: What is the difference between the binomial probability formula and the binomial cumulative distribution function?

A: The binomial probability formula calculates the probability of k successes in n trials, while the binomial cumulative distribution function calculates the probability of at most k successes in n trials.

Q: How do I calculate the probability of at least k successes?

A: To calculate the probability of at least k successes, you need to subtract the probability of at most k-1 successes from 1.

Q: What is the significance of the TI-84 Plus calculator in hypothesis testing and probability calculations?

A: The TI-84 Plus calculator is a powerful tool for performing various mathematical calculations, including hypothesis testing and probability calculations. It provides a user-friendly interface and accurate results, making it an essential tool for students and professionals in statistics and mathematics.

Q: Can I use the TI-84 Plus calculator for other types of probability calculations?

A: Yes, the TI-84 Plus calculator can be used for other types of probability calculations, such as:

  • Poisson distribution
  • Normal distribution
  • Exponential distribution
  • Chi-squared distribution

Q: How do I troubleshoot common errors on the TI-84 Plus calculator?

A: Common errors on the TI-84 Plus calculator include:

  • Incorrect entry of values
  • Incorrect selection of functions
  • Incorrect calculation of probabilities

To troubleshoot these errors, follow these steps:

  1. Check the entry of values for accuracy.
  2. Verify the selection of functions.
  3. Recalculate the probabilities using the correct formula.

Q: Can I use the TI-84 Plus calculator for other types of mathematical calculations?

A: Yes, the TI-84 Plus calculator can be used for other types of mathematical calculations, such as:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus

Q: How do I update the TI-84 Plus calculator software?

A: To update the TI-84 Plus calculator software, follow these steps:

  1. Connect the calculator to a computer using a USB cable.
  2. Download the latest software from the Texas Instruments website.
  3. Install the software on the calculator.

Q: Can I use the TI-84 Plus calculator for educational purposes?

A: Yes, the TI-84 Plus calculator is widely used in educational institutions for teaching statistics and mathematics. It provides a user-friendly interface and accurate results, making it an essential tool for students and teachers.

Q: How do I get support for the TI-84 Plus calculator?

A: To get support for the TI-84 Plus calculator, follow these steps:

  1. Visit the Texas Instruments website.
  2. Contact the customer support team.
  3. Check the user manual and online resources for troubleshooting and FAQs.

By following these FAQs, you can effectively use the TI-84 Plus calculator for hypothesis testing and probability calculations, and troubleshoot common errors.