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Chapter 6: Problem 7

How do the width and height of a normal distribution change when its mean remains the same but its standard deviation decreases?

Short Answer

Expert verified
When the standard deviation of a normal distribution decreases while the mean remains the same, the distribution becomes narrower (decrease in width) and taller (increase in height).

Step by step solution

01

Understand Normal Distribution

A Normal Distribution is defined by two parameters: the mean (碌), which determines the center of the distribution, and the standard deviation (蟽), which determines the width and the height of the distribution.
02

Effect of Decreasing Standard Deviation on Width

If the standard deviation decreases, the values in the distribution are more concentrated around the mean. This means that the width of the distribution decreases. In other words, the distribution curve becomes narrower.
03

Effect of Decreasing Standard Deviation on Height

Since the area under the curve must remain 1 (the probability of all possible outcomes), when the standard deviation decreases and the curve becomes narrower, it also becomes taller to compensate for the reduced width.

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Most popular questions from this chapter

For the standard normal distribution, find the area within one standard deviation of the mean - that is, the area between \(\mu-\sigma\) and \(\mu+\sigma .\)

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour

Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by this company malfunction within a 2-year period. The company recently mailed 500 such calculators to its customers. a. Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2 -year period. b. What is the probability that 27 or more of the 500 calculators will be returned for refund or replacement within a 2 -year period? c. What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2 -year period?

The amount of time taken by a bank teller to serve a randomly selected customer has a normal distribution with a mean of 2 minutes and a standard deviation of \(.5\) minute. a. What is the probability that both of two randomly selected customers will take less than 1 minute each to be served? b. What is the probability that at least one of four randomly selected customers will need more than \(2.25\) minutes to be served?

The average monthly mortgage payment for all homeowners in a city is $$\$ 2850.$$ Suppose that the distribution of monthly mortgages paid by homeowners in this city follow an approximate normal distribution with a mean of $$\$ 2850$$ and a standard deviation of $$\$ 420.$$ Find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is a. less than $$\$ 1200$$ b. between $$\$ 2300$$ and $$\$ 3140$$ c. more than $$\$ 3600$$ d. between $$\$ 3200$$ and $$\$ 3700$$
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