The time spent (in days) waiting for a heart transplant in two states for patients with type A+ blood can be approximated by a normal distribution, as shown in the graph to the right. mean of =128 Standard deviation of=23.3

To find the probability of waiting a certain number of days for a heart transplant, we can use the properties of the normal distribution. In this case, the mean (μ) is 128 and the standard deviation (σ) is 23.3.

The normal distribution is symmetrical, so we can use the properties of the standard normal distribution when the data is standardized. To standardize a value, we subtract the mean and divide by the standard deviation.

Let's say we want to find the probability of waiting less than or equal to 100 days for a heart transplant. We need to standardize the value of 100.

Z = (X - μ) / σ,
where X is the value we want to standardize, μ is the mean, and σ is the standard deviation.

Z = (100 - 128) / 23.3 = -1.2

Now, we can use a standard normal distribution table or a calculator to find the probability associated with a Z-score of -1.2. For example, looking up the Z-score of -1.2 in the table, we find that the probability is approximately 0.1151, which means there is a 11.51% chance of waiting less than or equal to 100 days for a heart transplant in these two states.

Similarly, you can find the probabilities for other waiting times by standardizing the values and looking up the corresponding probabilities in the standard normal distribution table or using a calculator.

To find the probability of waiting for a heart transplant within a certain number of days, we can use the standard normal distribution.

Step 1: Standardize the value using the z-score formula:
z = (x - mean) / standard deviation

Step 2: Calculate the z-score for the given situation:
For the first state:
z = (x - 128) / 23.3

For the second state, if the mean and standard deviation are the same:
z = (x - 128) / 23.3

Step 3: Use a standard normal distribution table or calculator to find the probability corresponding to the z-score obtained.

For example, to find the probability of waiting less than a certain number of days, use the cumulative distribution function (CDF) on the standard normal distribution.

Let's say we want to find the probability of waiting less than 110 days for a heart transplant in the first state.

Step 4: Calculate the z-score for the desired value of 110:
z = (110 - 128) / 23.3 = -0.773

Step 5: Use a standard normal distribution table, calculator, or software to find the probability corresponding to the z-score of -0.773. The probability will represent the area under the curve to the left of the z-score.

If you need assistance with specific calculations or have any further questions, please let me know!