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Chapter 6: Problem 8

Do the width and/or height of a normal distribution change when its standard deviation remains the same but its mean increases?

Short Answer

Expert verified
No, the width and height of a normal distribution do not change when its mean increases but its standard deviation remains the same. Only the position of the distribution shifts.

Step by step solution

01

Understanding of concepts involved

The mean of a normal distribution tells where the center of the distribution is. The standard deviation shows how wide the spread of the data in the distribution. A change in the mean while keeping the standard deviation constant would result in a shift in the position of the distribution, either to the right (for an increase in mean) or left (for a decrease in mean). It doesn't affect the width or the height of the distribution.
02

Analyzing the impact of a change in mean

When the mean of a distribution increases while the standard deviation remains the same, the peak of the distribution shifts to the right. No change occurs in the height or width of the distribution. The peak of the distribution is still at the same height, the tails still extend the same distance on both sides, and the area under the curve stays the same too.
03

Conclusion

So, a change in the mean of a distribution, while keeping the standard deviation constant, only affects the position of the distribution (shifts it to the right or left). There are no alterations to the width or height of the distribution.

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Most popular questions from this chapter

Find the following binomial probabilities using the normal approximation. a. \(n=70, \quad p=.30, \quad P(x=18)\) b. \(n=200, \quad p=.70, \quad P(133 \leq x \leq 145)\) c. \(n=85, \quad p=.40, \quad P(x \geq 30)\) d. \(n=150, \quad p=.38, \quad P(x \leq 62)\)

Find the following probabilities for the standard normal distribution. a. \(P(z<-1.31)\) b. \(P(1.23 \leq z \leq 2.89)\) c. \(P(-2.24 \leq z \leq-1.19)\) d. \(P(z<2.02)\)

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For the standard normal distribution, what is the area within three standard deviations of the mean?

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