Questions: The total area under a normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 7.5. Find the mean of the first distribution.

The total area under a normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 7.5. Find the mean of the first distribution.

Transcript text: The total area under a normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 12 is three times the area under normal distribution curve with a standard deviation of 7.5. Find the mean of the first distribution.

Solution

Solution Steps

Step 1: Verify the Line of Symmetry

The line of symmetry for a standard normal distribution is given by \( x = 0 \). This means that the distribution is symmetric around the mean, which is equal to zero in the case of a standard normal distribution.

Step 2: Evaluate the Area Under the Curve

The statement that the total area under a normal distribution curve with a standard deviation of \( 12 \) is three times the area under a normal distribution curve with a standard deviation of \( 7.5 \) is false. The total area under any normal distribution curve is always \( 1 \), regardless of the standard deviation. Therefore, this statement does not hold true.

Step 3: Assess the Symmetry of Normal Distributions

The assertion that normal distributions are bell-shaped but do not have to be symmetric is also false. In fact, normal distributions are always symmetric about their mean.

Step 4: Identify the Inflection Points

The statement regarding the inflection points of any normal distribution is true. The inflection points occur at one standard deviation on either side of the mean, specifically at \( \mu - \sigma \) and \( \mu + \sigma \).