Question

Calculate the area under the standard normal curve to the left of these values: a. \(z=-.90\) b. \(z=2.34\) c. \(z=5.4\)

Short answer

Answer: a. 18.41%, b. 99.06%, and c. approximately 100%.

Step-by-step solution

Refer to the standard normal table or use a calculator/software

Find a standard normal table (also called the \(z\)-table) or use a calculator or software that provides the area to the left of the given \(z\)-score. Remember, the value from the table or calculator will give you the probability.

Identify the \(z\)-score and look up or calculate the area to the left

For each of the given \(z\)-scores, locate the value in the \(z\)-table or calculate the area to the left of the particular value using a calculator or software. a. \(z=-.90\) Using a standard normal table, the area to the left of the \(z\)-score \(-0.90\) is 0.1841. b. \(z=2.34\) Look up the \(z\)-score \(2.34\) in the \(z\)-table or calculate the area to the left using calculator/software; the value is 0.9906. c. \(z=5.4\) The \(z\)-table may not display values as high as \(5.4\). However, you could use a calculator or software to find the area to the left of the \(z\)-score \(5.4\). The value is approximately 1.0000 since the area under the curve is essentially the entire distribution at this point.

Interpret the results

Now that we have the area under the curve to the left of each \(z\)-score, we can interpret the results: a. For \(z=-0.90\), there is an 18.41% (0.1841) chance that a value from the standard normal distribution will be less than or equal to \(-0.90\). b. For \(z=2.34\), there is a 99.06% (0.9906) chance that a value from the standard normal distribution will be less than or equal to \(2.34\). c. For \(z=5.4\), there is essentially a 100% (1.0000) chance that a value from the standard normal distribution will be less than or equal to \(5.4\).