/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Home runs Based on data from the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 37

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 .

Short Answer

Expert verified
a. No, likely Poisson; b. Normal shape, mean 1.0, SD 0.0781; c. Probability is approximately 0.

Step by step solution

01

Assess Normality of Distribution

The mean and standard deviation of the number of home runs the San Francisco Giants hits per game (X) are both 1. However, because this is count data (home runs), it is more likely to follow a Poisson distribution rather than a normal distribution. Count data with relatively small means like this often do not distribute normally. Hence, it's unlikely that \(X\) is normally distributed.
02

Characteristics of the Sampling Distribution

According to the Central Limit Theorem, regardless of the shape of the population distribution, the sampling distribution of the sample mean will be approximately normal if the sample size (in this case, 162 games) is large. Therefore, the shape will be approximately normal. The mean of the sampling distribution will be equal to the mean of \(X\), which is 1.0. The standard deviation (standard error) of the sampling distribution is calculated using the formula: \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma = 1.0 \) and \( n = 162 \). This equals \( \frac{1}{\sqrt{162}} \approx 0.0781 \).
03

Calculate Probability of Exceeding 1.50

To find the probability that the mean number of home runs per game exceeds 1.50, we first find the z-score using the formula \( z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \), where \( \bar{x} = 1.50 \), \( \mu_{\bar{x}} = 1.0 \), and \( \sigma_{\bar{x}} = 0.0781 \): \[ z = \frac{1.50 - 1.0}{0.0781} \approx 6.4 \]. Then, using a z-table or normal distribution calculator, we find the probability of \( z > 6.4 \). Since 6.4 is much larger than typical z-scores on a standard normal curve, the probability is essentially 0.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Normal Distribution
The concept of a normal distribution is central to understanding data distribution in statistics. A normal distribution is often referred to as a bell curve due to its symmetric, bell-shaped appearance. In a normal distribution, most of the data values are clustered around the mean, with progressively fewer values appearing as you move away from the mean. This makes the mean, median, and mode identical.

However, not all datasets or types of data follow a normal distribution. For example, in the case of the San Francisco Giants' home runs per game, the distribution of this count data is unlikely to be normal. Instead, this kind of data, where events occur independently with a certain mean rate, usually follows a Poisson distribution.
  • Normal distributions have bell-shaped curves.
  • Mean, median, and mode are equal in a normal distribution.
  • Normal distributions are not ideal for count data like home runs, which may follow a Poisson distribution instead.
Central Limit Theorem
The Central Limit Theorem (CLT) is a pivotal concept in statistics. It states that the sampling distribution of the sample means will tend to be normal regardless of the population's distribution, provided the sample size is large enough.

In the scenario of the average home runs hit by the Giants over a season of 162 games, the CLT suggests that, while individual games might not have home runs that follow a normal distribution, the average of those games will. This is because the CLT allows for approximate normality with large sample sizes such as 162 games. This is incredibly valuable for making inferences about the population from samples.

The sampling distribution will have the same mean as the population mean and a reduced standard deviation known as the standard error which is calculated by dividing the population standard deviation by the square root of the number of samples.
  • The CLT ensures approximate normality for the sample means.
  • This theorem is applicable despite the original distribution shape.
  • With a large sample size, like 162 games, the sample mean distribution becomes normal.
Poisson Distribution
The Poisson distribution is a type of discrete probability distribution best suited for count data or events that occur independently over a constant mean rate. It's ideal for situations such as the number of home runs scored in a game by the San Francisco Giants, especially when the mean is low.

The shape of a Poisson distribution can vary depending on its mean. With a small mean, the distribution is skewed to the right, meaning there is a higher probability of observing fewer occurrences. As the mean increases, the distribution becomes more symmetric and starts to resemble a normal distribution.

Key characteristics of a Poisson distribution include:
  • Event occurrences are independent.
  • Mean rate is constant over a given interval.
  • Skewness decreases as the mean increases, gradually mimicking a normal distribution.
Overall, Poisson distribution helps model scenarios where the mean is small, and provides insights that can sometimes contradict assumptions of normality in the dataset.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Survey accuracy A study investigating the relationship between age and annual medical expenses randomly samples 100 individuals in Davis, California. It is hoped that the sample will have a similar mean age as the entire population. a. If the standard deviation of the ages of all individuals in Davis is \(\sigma=15,\) find the probability that the mean age of the individuals sampled is within two years of the mean age for all individuals in Davis. (Hint: Find the sampling distribution of the sample mean age and use the central limit theorem. You don't have to know the population mean to answer this, but if it makes it easier, use a value such as \(\mu=30 .\) ) b. Would the probability be larger, or smaller, if \(\sigma=10 ?\) Why?

Basketball shooting In college basketball, a shot made from beyond a designated arc radiating about 20 feet from the basket is worth three points, instead of the usual two points given for shots made inside that arc. Over his career, University of Florida basketball player Lee Humphrey made \(45 \%\) of his three-point attempts. In one game in his final season, he made only 3 of 12 three-point shots, leading a TV basketball analyst to announce that Humphrey was in a shooting slump. a. Assuming Humphrey has a \(45 \%\) chance of making any particular three-point shot, find the mean and standard deviation of the sampling distribution of the proportion of three-point shots he will make out of 12 shots. b. How many standard deviations from the mean is this game's result of making 3 of 12 three-point shots? c. If Humphrey was actually not in a slump but still had a \(45 \%\) chance of making any particular three-point shot, explain why it would not be especially surprising for him to make only 3 of 12 shots. Thus, this is not really evidence of a shooting slump.

Baseball hitting Suppose a baseball player has a 0.200 probability of getting a hit in each time at-bat. a. Describe the shape, mean, and standard deviation of the sampling distribution of the proportion of times the player gets a hit after 36 at- bats. b. Explain why it would not be surprising if the player has a 0.300 batting average after 36 at-bats.

What good is a standard deviation? Explain how the standard deviation of the sampling distribution of a sample proportion gives you useful information to help gauge how close a sample proportion falls to the unknown population proportion.

Multiple choice: CLT The central limit theorem implies a. All variables have approximately bell-shaped data distributions if a random sample contains at least about 30 observations. b. Population distributions are normal whenever the population size is large. c. For sufficiently large random samples, the sampling distribution of \(\bar{x}\) is approximately normal, regardless of the shape of the population distribution. d. The sampling distribution of the sample mean looks more like the population distribution as the sample size increases.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.