Find the area under the standard normal curve to the left of
a. z = 2.56
b. z = 0.53
c. z = -0.94
d. z = -1.30
The correct answer and explanation is:
Here are the areas under the standard normal curve to the left of each given z-value:
a. For z = 2.56
Area to the left ≈ 0.9948
b. For z = 0.53
Area to the left ≈ 0.7019
c. For z = -0.94
Area to the left ≈ 0.1736
d. For z = -1.30
Area to the left ≈ 0.0968
Explanation:
The standard normal distribution is a bell-shaped curve with a mean of 0 and a standard deviation of 1. The total area under the curve is equal to 1, which corresponds to 100% probability. The area to the left of a specific z-value represents the cumulative probability that a randomly selected value from the distribution is less than that z-value.
To find these areas, a standard normal table or a calculator with the cumulative distribution function (CDF) for the standard normal distribution is used. This function gives the probability that a value is less than or equal to a given z-score.
For positive z-values (like 2.56 and 0.53), the area to the left is greater than 0.5 because the mean (0) divides the distribution into two equal halves. As the z-value increases positively, the cumulative area approaches 1.
For negative z-values (like -0.94 and -1.30), the area to the left is less than 0.5. This is because these values lie to the left of the mean, so the cumulative probability is smaller. As the z-value becomes more negative, the cumulative area approaches 0.
The exact values come from either a table or software that calculates the integral of the probability density function from negative infinity up to the z-value. This is important for many statistical analyses, including hypothesis testing and confidence interval construction.
In summary, the area to the left of a z-score tells the likelihood that a standard normal random variable is less than that z-score. These values are essential for interpreting probabilities and making decisions based on normal distributions.