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Chapter 7: Problem 39

Bank machine withdrawals An executive in an Australian savings bank decides to estimate the mean amount of money withdrawn in bank machine transactions. From past experience, she believes that \(\$ 50\) (Australian) is a reasonable guess for the standard deviation of the distribution of withdrawals. She would like her sample mean to be within \(\$ 10\) of the population mean. Estimate the probability that this happens if she randomly samples 100 withdrawals. (Hint: Find the standard deviation of the sample mean. How many standard deviations does \(\$ 10\) equal?)

Short Answer

Expert verified
The probability is approximately 95.45% that the sample mean is within \( \$10 \) of the population mean.

Step by step solution

01

Understanding the Problem

We need to estimate the probability that the sample mean of withdrawals is within \( \\(10 \) of the population mean while knowing the standard deviation of the withdrawal distribution is \( \\)50 \). Also, the sample size is 100 withdrawals.
02

Calculate Standard Error of the Sample Mean

The standard error (SE) of the sample mean is given by the formula \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size. Plugging in the values: \( SE = \frac{50}{\sqrt{100}} = \frac{50}{10} = 5 \).
03

Determine the Number of Standard Deviations for \( \$10 \)

To find out how many standard deviations \( \\(10 \) is from the sample mean, divide \( \\)10 \) by the standard error: \( \text{Number of SDs} = \frac{10}{5} = 2 \).
04

Calculate the Probability Using the Standard Normal Distribution

The probability that the sample mean is within 2 standard deviations of the population mean involves looking up the standard normal distribution. The probability that a standard normal random variable is within 2 standard deviations of the mean is about 0.9545, according to standard normal distribution tables.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
The standard deviation is a measure of how spread out numbers are in a data set. It tells us how much the values in a data set deviate from the average. In the context of bank withdrawals, a standard deviation of \(50\) means that the amounts withdrawn typically vary by about \(50\) from the average withdrawal amount.
  • It helps in understanding the variability of data.
  • In our exercise, knowing the standard deviation allows us to see how much individual withdrawals might differ from the mean.
In statistical terms, if the standard deviation is large, the data points are spread out widely. If it's small, they are clustered closely around the mean.
Sample Mean
The sample mean is the average of all values in a sample. This average is crucial when you only have a subset of data from a larger population. In our exercise, the sample mean is the average withdrawal amount calculated from 100 transactions. This sample mean estimates the population mean, which is the average amount people withdraw from the bank in general.
  • The sample mean helps provide an indication of the central tendency of the data.
  • It acts as a good representation of the overall population if the sample is random and unbiased.
Thus, the sample mean can help in making informed guesses about the population's average withdrawal amount.
Standard Error
The standard error (SE) of the sample mean is a statistical term that measures the accuracy with which a sample represents a population. It's calculated by dividing the population standard deviation by the square root of the sample size.For example:\[ SE = \frac{50}{\sqrt{100}} = 5 \]This tells us how much the sample mean would fluctuate from the actual population mean if different samples were taken.
  • A smaller standard error indicates more reliable and precise estimates of the population mean.
  • It helps gauge the reliability of the sample mean as a reflection of the population mean.
Normal Distribution
The normal distribution is a bell-shaped curve that is symmetrical around the mean. It describes how the values of a variable are distributed. In many cases, especially with large sample sizes, data tends to follow this pattern. In the bank withdrawal example, if the sample mean follows a normal distribution, we can apply properties of the normal distribution to determine probabilities, like finding how likely our sample mean is within a certain range of the population mean.
  • This distribution is central to statistical analysis and probability estimations.
  • It allows us to use standard deviation to make probability statements about how data behaves.
When we say that the sample mean is within 2 standard deviations, the normal distribution tells us the probability of this occurring, which in our case is about 0.9545.

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Most popular questions from this chapter

PDI The scores on the Psychomotor Development Index (PDI), a scale of infant development, have a normal population distribution with mean 100 and standard deviation 15. An infant is selected at random. a. Find the \(z\) -score for a PDI value of 90 . b. A study uses a random sample of 225 infants. Using the sampling distribution of the sample mean PDI, find the \(z\) -score corresponding to a sample mean of 90 . c. Explain why a PDI value of 90 is not surprising, but a sample mean PDI score of 90 for 225 infants would be surprising.

Blood pressure Vincenzo Baranello was diagnosed with high blood pressure. He was able to keep his blood pressure in control for several months by taking blood pressure medicine (amlodipine besylate). Baranello's blood pressure is monitored by taking three readings a day, in early morning, at mid-day, and in the evening. a. During this period, the probability distribution of his systolic blood pressure reading had a mean of 130 and a standard deviation of \(6 .\) If the successive observations behave like a random sample from this distribution, find the mean and standard deviation of the sampling distribution of the sample mean for the three observations each day. b. Suppose that the probability distribution of his blood pressure reading is normal. What is the shape of the sampling distribution? Why? c. Refer to part b. Find the probability that the sample mean exceeds \(140,\) which is considered problematically high. (Hint: Use the sampling distribution, not the probability distribution for each observation.)

Home runs Based on data from the 2010 major league baseball season, \(X=\) number of home runs the San Francisco Giants hits in a game has a mean of 1.0 and a standard deviation of 1.0 . a. Do you think that \(X\) has a normal distribution? Why or why not? b. Suppose that this year \(X\) has the same distribution. Report the shape, mean, and standard deviation of the sampling distribution of the mean number of home runs the team will hit in its 162 games. c. Based on the answer to part b, find the probability that the mean number of home runs per game in this coming season will exceed 1.50 .

True or false \(\quad\) As the sample size increases, the standard deviation of the sampling distribution of \(\bar{x}\) increases. Explain your answer.

Playing roulette \(\quad\) A roulette wheel in Las Vegas has 38 slots. If you bet a dollar on a particular number, you'll win \(\$ 35\) if the ball ends up in that slot and \(\$ 0\) otherwise. Roulette wheels are calibrated so that each outcome is equally likely. a. Let \(X\) denote your winnings when you play once. State the probability distribution of \(X\). (This also represents the population distribution you would get if you could play roulette an infinite number of times.) It has mean 0.921 and standard deviation 5.603 . b. You decide to play once a minute for 12 hours a day for the next week, a total of 5040 times. Show that the sampling distribution of your sample mean winnings has mean \(=0.921\) and standard deviation \(=0.079 .\) c. Refer to part b. Using the central limit theorem, find the probability that with this amount of roulette playing, your mean winnings is at least \(\$ 1,\) so that you have not lost money after this week of playing. (Hint: Find the probability that a normal random variable with mean 0.921 and standard deviation 0.079 exceeds \(1.0 .\) )
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