What is the probability that a normally distributed random variable lies within 1.65 standard deviations of the mean?Note: You can use the Z table here.95%90%85%80%

The probability that a normally distributed random variable lies within 1.65 standard deviations of the mean can be found using the Z-table.

Step 1: Identify the Z-score. In this case, it is 1.65.

Step 2: Look up the Z-score in the Z-table. The value in the Z-table for 1.65 is 0.9505. This means that 95.05% of the data falls within 1.65 standard deviations of the mean on one side of the mean.

Step 3: Since a normal distribution is symmetrical, the probability that a normally distributed random variable lies within 1.65 standard deviations of the mean on either side is 2*0.9505 = 1.901 or 190.1%.

However, probabilities cannot exceed 100%. This discrepancy arises because the Z-table gives the probability up to the Z-score from the mean, not between the mean and the Z-score.

Step 4: To find the probability that a normally distributed random variable lies within 1.65 standard deviations of the mean, we need to subtract the probability of the data falling below the mean (which is 0.5 or 50% for a normal distribution) from the value we obtained from the Z-table.

So, 0.9505 - 0.5 = 0.4505 or 45.05%.

Step 5: Since the distribution is symmetrical, the probability that a normally distributed random variable lies within 1.65 standard deviations of the mean on either side is 2*0.4505 = 0.901 or 90.1%.

So, the answer is approximately 90%.