Question

N a certain store, there is a .03 probability that the scanned price in the bar code scanner will not match the advertised price. the cashier scans 780 items. (a-1) what is the expected number of mismatches? (round your answer to the nearest whole number.) (a-2) what is the standard deviation? (use your rounded number for the expected number of mismatches for the calculation of standard deviation. round your final answer to 4 decimal places.) (b) what is the approximate normal probability of at least 18 mismatches? (round the z-value to 2 decimal places. use appendix c-2 to find probabilities. round your final answer to 4 decimal places.) (c) what is the approximate normal probability of more than 32 mismatches? (round the z-value to 2 decimal places. use appendix c-2 to find probabilities. round your final answer to 4 decimal places.)

Answer 1

The expected number of mismatches is 23, and the standard deviation is 4.7232. To determine the probabilities of at least 18 mismatches or more than 32 mismatches, we use the normal approximation to the binomial distribution and look up the corresponding areas on the standard normal distribution table.

Step-by-step explanation:

The question deals with the probability of price mismatches when a cashier scans items at a store, and it requires the application of statistical methods to calculate expected values, standard deviations, and probabilities.

(a-1) Expected Number of Mismatches

To calculate the expected number of mismatches, we multiply the probability of a mismatch (0.03) by the total number of items scanned (780):

Expected number of mismatches = 0.03 × 780 = 23.4

When rounded to the nearest whole number, the expected number of mismatches is 23.

(a-2) Standard Deviation

The standard deviation for a binomial distribution is calculated using the formula √(np(1-p)). Since we are asked to use the rounded expected number for this calculation, we will use 23 as 'np'.

Standard deviation = √(23(1-0.03)) = √(22.31) = 4.7232 (rounded to four decimal places)

(b) Probability of At Least 18 Mismatches

To find the normal probability of at least 18 mismatches, we use the normal approximation to the binomial. First, we calculate the z-score:

Z = (x - μ) / σ = (18 - 23) / 4.7232 = -1.0577 (rounded to two decimal places)

Consulting the z-table, we can find P(Z ≥ -1.0577). Since the z-table gives the area to the left of the z-score, we need 1 - P(Z < -1.0577) to find the probability of at least 18 mismatches.

(c) Probability of More Than 32 Mismatches

Similarly, to find the probability of more than 32 mismatches, we calculate the z-score for 32 and use the z-table. Subtracting from 1 gives us the probability of more than x mismatches.