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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

The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $$\$ 95$$ and a standard deviation of $$\$ 20.$$ If the management wants to give free gifts to at most \(10 \%\) of the customers, what should the amount be above which a customer would receive a free gift?

Two companies, A and B, drill wells in a rural area. Company A charges a flat fee of \(\$ 3500\) to drill a well regardless of its depth. Company B charges \(\$ 1000\) plus \(\$ 12\) per foot to drill a well. The depths of wells drilled in this area have a normal distribution with a mean of 250 feet and a standard deviation of 40 feet. a. What is the probability that Company B would charge more than Company A to drill a well? b. Find the mean amount charged by Company B to drill a well.

According to the records of an electric company serving the Boston area, the mean electricity consumption for all households during winter is 1650 kilowatt-hours per month. Assume that the monthly electricity consumptions during winter by all households in this area have a normal distribution with a mean of 1650 kilowatt-hours and a standard deviation of 320 kilowatt-hours. a. Find the probability that the monthly electricity consumption during winter by a randomly selected household from this area is less than 1950 kilowatt- hours. b. What percentage of the households in this area have a monthly electricity consumption of 900 to 1300 kilowatt-hours?

Jenn Bard, who lives in the San Francisco Bay area, commutes by car from home to work. She knows that it takes her an average of 28 minutes for this commute in the morning. However, due to the variability in the traffic situation every morning, the standard deviation of these commutes is 5 minutes. Suppose the population of her morning commute times has a normal distribution with a mean of 28 minutes and a standard deviation of 5 minutes. Jenn has to be at work by \(8: 30\) A.M. every morning. By what time must she leave home in the morning so that she is late for work at most \(1 \%\) of the time?

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?
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