**Answer:**

**a) **There is a 69.155 probability that a randomly-selected customer will take less than 3 minutes.

**b) **There is a 76.11% probability that the average time of two randomly-selected customers will take less than 3 minutes.

**c) **There is a 90.82% probability that the average time of 64 randomly-selected customers will take less than 2.8 minute.

**d)** There is a 93.19% probability that the average time of 81 randomly-selected customers will take less than 2.8 minutes.

**e) **There is a 99.38% probability that the average time of 225 randomly-selected customers will take less than 2.8 minutes.

**f) **There is a 99.95% probability that the average time of 400 randomly-selected customers will take less than 2.8 minutes.

**Step-by-step explanation:**

**Problems of normally distributed samples can be solved using the z-score formula.**

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

**In this problem, we have that:**

The mean checkout time in the express lane of a large grocery store is 2.7 minutes, and the standard deviation is 0.6 minutes. This means that .

**(a) What is the probability that a randomly-selected customer will take less than 3 minutes?**

This is the pvalue of Z when . So:

has a pvalue of 0.6915.

There is a 69.155 probability that a randomly-selected customer will take less than 3 minutes.

**(b) What is the probability that the average time of two randomly-selected customers will take less than 3 minutes?**

We are now working with the average of a sample, so we must use instead of in the formula of Z.

has a pvalue f 0.7611.

There is a 76.11% probability that the average time of two randomly-selected customers will take less than 3 minutes.

**(c) The probability that the average time of 64 randomly-selected customers will take less than 2.8 minutes is**

This is the pvalue of Z when

has a pvalue of 0.9082.

There is a 90.82% probability that the average time of 64 randomly-selected customers will take less than 2.8 minute.

**(d) The probability that the average time of 81 randomly-selected customers will take less than 2.8 minutes is.**

has a pvalue of 0.9319.

There is a 93.19% probability that the average time of 81 randomly-selected customers will take less than 2.8 minutes.

**(e) The probability that the average time of 225 randomly-selected customers will take less than 2.8 minutes is.**

has a pvalue of 0.9938.

There is a 99.38% probability that the average time of 225 randomly-selected customers will take less than 2.8 minutes.

**(f) The probability that the average time of 400 randomly-selected customers will take less than 2.8 minutes is.**

has a pvalue of 0.9995.

There is a 99.95% probability that the average time of 400 randomly-selected customers will take less than 2.8 minutes.