If np is more or equal to 5 and nq is more or equal to 5, estimate P(fewer than 6) with n = 14 and p = 0.5 by using the normal distribution as an approximation to the binomial distribution; if np < 5 or nq < 5, then state that the normal approximation is not suitable.
Could I get some help here,I d not get this kind of math.
a caterer os planning a large dinner party for 273 guests. Each table must seat and equal number of guests, more than 1 guest and fewer than 20. Determine the number of possible table sizes. If the caterers use 11 person tables, how many tables will they need?
Certainly! I can help you understand this math problem.
This problem involves using the normal distribution as an approximation to the binomial distribution. The binomial distribution describes the probability of getting a certain number of successes in a fixed number of independent Bernoulli trials.
To estimate P(fewer than 6), we need to first check if the conditions for using the normal approximation are met. The conditions state that np and nq should both be greater than or equal to 5.
Let's apply these conditions to the given problem:
- n = 14 (the fixed number of trials).
- p = 0.5 (the probability of success in each trial).
Now, we can calculate np and nq:
np = n * p = 14 * 0.5 = 7
nq = n * (1 - p) = 14 * (1 - 0.5) = 7
Since both np and nq are equal to 7, which is greater than or equal to 5, we can proceed with using the normal approximation.
To estimate P(fewer than 6), we need to find the z-score of the value 6 and then look up the corresponding probability in the standard normal table. The z-score is calculated using the formula: z = (x - np) / sqrt(npq).
In this case, x is 6, np is 7, and npq is calculated as np * (1 - p) = 7 * 0.5 = 3.5.
So, the z-score is: z = (6 - 7) / sqrt(3.5) ≈ -0.47
Looking up the corresponding probability in the standard normal table for a z-score of -0.47, we find the value to be approximately 0.3192.
Therefore, the estimated probability of getting fewer than 6 successes is approximately 0.3192.
I hope this explanation helps clarify how to approach this type of math problem. If you have any further questions, please let me know!