Approximately 10% Of American High School Students Drop Out ✓ Solved

Approximately 10 of American high school students drop out before graduation

Approximately 10% of American high school students drop out before graduation

Analyze the probability that all 20 students selected at random in a sample of American high school students, given that approximately 10% of these students drop out before graduation, have stayed in school and graduated. Show your work using an appropriate calculator function for probabilities involving binomial distributions.

Sample Paper For Above instruction

In this problem, we're asked to find the probability that all 20 selected students have graduated from high school, given that about 10% of students drop out before graduation. This is a classic binomial probability problem where the probability of success (a student graduating) is p = 0.9, since 10% drop out and 90% graduate.

Step 1: Define the problem parameters

  • Number of trials (students selected): n = 20
  • Probability of success (student graduates): p = 0.9
  • Number of successes desired: k = 20 (all students graduated)

Step 2: Recognize the binomial distribution

The probability of exactly k successes in n trials is given by the binomial probability mass function (PMF):

P(X = k) = C(n, k) p^k (1 - p)^{n - k}

However, because we want all 20 students to have graduated, we're calculating P(X = 20).

Step 3: Calculate the probability

Using a calculator's binomial probability function, such as BINOM.DIST in Excel, or a graphing calculator's binomial function, we input:

  • x = 20
  • n = 20
  • p = 0.9
  • cumulative = FALSE

Thus, P(all 20 students graduated) = BINOM.DIST(20, 20, 0.9, FALSE).

Step 4: Calculation result

Calculating this yields:

P = (0.9)^20 ≈ 0.1216

This indicates that there is approximately a 12.16% chance that all 20 students in the sample have graduated, assuming the dropout rate is 10% and students' successes are independent.

Conclusion: The probability that all 20 students selected at random have graduated is approximately 0.1216 using the binomial distribution model with p = 0.9.

References

  • Devore, J. L. (2015).Probability and Statistics for Engineering and the Sciences. Brooks/Cole.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012).Introduction to the Practice of Statistics. W.H. Freeman.
  • Applebaum, D. (2008). Probability and Statistics. Cengage Learning.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Ross, S. M. (2014). A First Course in Probability. Pearson.
  • Johnson, R. A., & Bhattacharyya, G. K. (2010). Introduction to Probability and Statistics. Wiley.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Ott, L. (2012). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
  • Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.

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