/*! 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 149 The dataset BaseballHits gives 2... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 149

The dataset BaseballHits gives 2010 season statistics for all Major League Baseball teams. We treat this as a sample of all MLB teams in all years. Computer output of descriptive statistics for the variable giving the batting average is shown: $$ \begin{aligned} &\text { Descriptive Statistics: BattingAvg }\\\ &\begin{array}{lrrrrr} \text { Variable } & \mathrm{N} & \mathrm{N}^{*} & \text { Mean } & \text { SE Mean } & \text { StDev } \\ \text { BattingAvg } & 30 & 0 & 0.25727 & 0.00190 & 0.01039 \\ \text { Minimum } & & \text { Q1 } & \text { Median } & \text { Q3 } & \text { Maximum } \\ 0.23600 & 0.24800 & 0.25700 & 0.26725 & 0.27600 \end{array} \end{aligned} $$ (a) How many teams are included in the dataset? What is the mean batting average? What is the standard deviation? (b) Use the descriptive statistics above to conduct a hypothesis test to determine whether there is evidence that average team batting average is different from \(0.250 .\) Show all details of the test. (c) Compare the test statistic and p-value you found in part (b) to the computer output below for the same data:

Short Answer

Expert verified
The number of teams included in the dataset is 30. The mean batting average is 0.25727, and the standard deviation is 0.01039. To test the hypothesis that the average team batting average is different from 0.250, we need to conduct a t-test which will provide a t-value and p-value. These values need to be compared with the respective outputs provided by the computer to finalize our results.

Step by step solution

01

Interpret the Descriptive Statistics

From the data we can see that \( N \) denotes the number of data points, hence the number of teams included in the dataset is 30. The mean (average) batting average is given as 0.25727, and the standard deviation is listed as 0.01039.
02

Conduct the Hypothesis Test

A hypothesis test is a statistical test that is used to determine whether there is enough evidence to reject a null hypothesis (\( H_0 \)). In this case, the null hypothesis (\( H_0 \)) is that the mean batting average for the population is \(0.250\), and the alternative hypothesis (\( H_a \)) is that the mean batting average is not \(0.250\). We use the following formula for the test statistic, \( t \): \( t = \frac{X - \mu}{s / \sqrt{N}} \) where \( X \) is the sample mean, \( \mu \) is the population mean under the null hypothesis, \( s \) is the sample standard deviation, and \( N \) is the size of the sample. Substituting the values we have, \( t = \frac{0.25727 - 0.250}{0.01039 / \sqrt{30}} \)
03

Comparison of Test Results

By conducting the test we have calculated a t-value. This t-value and an associated p-value must be compared with the provided computer output. The p-value is calculated based on the t-value, then it can decide whether to reject the null hypothesis (\( H_0 \)) or not. If the p-value is less than the predetermined threshold (commonly 0.05), then there is a statistically significant difference, and the null hypothesis is rejected.

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Key Concepts

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

Descriptive Statistics
Descriptive statistics provide a way to summarize and describe the main features of a data set in quantitative terms. These statistics are incredibly useful for getting a quick insight into the general behavior of the data without making any assertions about the data causing certain effects.

For instance, in our example of Major League Baseball teams' batting averages, the descriptive statistics include the mean or average batting average, which is calculated as the sum of all batting averages divided by the number of teams. Here, the mean batting average is 0.25727. Descriptive statistics also outline the data's dispersion or spread, with measures like the minimum and maximum values, the first (Q1) and third quartiles (Q3), and the median 鈥� the middle value of the data set when ordered.

The dataset itself comes from 2010 season statistics and includes 30 Major League Baseball teams, which we treat as a sample of all MLB teams across all years. Descriptive statistics are fundamental as they set the stage for further statistical analysis, such as hypothesis testing, by providing a backbone of numerical data.
Standard Deviation
The standard deviation is a measurement of the amount of variation or dispersion in a set of values. In simpler terms, it tells us how much the individual data points differ from the mean of the data set.

In the context of the batting average statistics from our MLB teams example, the standard deviation is 0.01039. This signifies how far on average each team's batting average is from the mean batting average of 0.25727. A smaller standard deviation indicates that the values are closer to the mean (more consistency among team's batting averages), whereas a larger standard deviation would suggest a wider variation in the data (more variability among team's batting averages).

Understanding standard deviation helps us interpret the spread of data which in turn, is crucial when we perform hypothesis testing鈥攊t affects how we evaluate the variability of the sample in relation to our hypothesis.
P-value
In hypothesis testing, the p-value is a crucial statistic that indicates the probability of obtaining test results at least as extreme as the ones observed during the test, assuming that the null hypothesis is correct. It鈥檚 a measure that helps us determine whether to reject the null hypothesis.

For example, after calculating the test statistic for our MLB batting average hypothesis test, we look up or calculate the corresponding p-value. If this p-value is lower than our significance level (often set at 0.05), we would reject the null hypothesis, suggesting that there is a statistically significant difference from what was expected under that hypothesis. Conversely, a higher p-value would indicate that the observed data is consistent with the null hypothesis, and therefore, we would not have sufficient evidence to reject it.

The p-value is the bridge between the calculated statistics from our sample and the decisions we make regarding the entire population鈥攖he lower the p-value, the stronger the evidence against the null hypothesis. In the MLB example, we use the mean, standard deviation, and sample size to calculate the test statistic, which is then used to determine the p-value and make an inference about the population.

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Most popular questions from this chapter

Standard Error from a Formula and Simulation In Exercises 6.15 to \(6.18,\) find the mean and standard error of the sample proportions two ways: (a) Use StatKey or other technology to simulate at least 1000 sample proportions. Give the mean and standard error and comment on whether the distribution appears to be normal. (b) Use the formulas in the Central Limit Theorem to compute the mean and standard error. Are the results similar to those found in part (a)? Sample proportions of sample size \(n=50\) from a population with \(p=0.25\)

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A young statistics professor decided to give a quiz in class every week. He was not sure if the quiz should occur at the beginning of class when the students are fresh or at the end of class when they've gotten warmed up with some statistical thinking. Since he was teaching two sections of the same course that performed equally well on past quizzes, he decided to do an experiment. He randomly chose the first class to take the quiz during the second half of the class period (Late) and the other class took the same quiz at the beginning of their hour (Early). He put all of the grades into a data table and ran an analysis to give the results shown below. Use the information from the computer output to give the details of a test to see whether the mean grade depends on the timing of the quiz. (You should not do any computations. State the hypotheses based on the output, read the p-value off the output, and state the conclusion in context.) $$ \begin{aligned} &\text { Two-Sample T-Test and Cl }\\\ &\begin{array}{lrrrr} \text { Sample } & \mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ \text { Late } & 32 & 22.56 & 5.13 & 0.91 \\ \text { Early } & 30 & 19.73 & 6.61 & 1.2 \end{array} \end{aligned} $$ Difference \(=\mathrm{mu}\) (Late) \(-\mathrm{mu}\) (Early) Estimate for difference: 2.83 \(95 \%\) Cl for difference: (-0.20,5.86) T-Test of difference \(=0\) (vs not \(=\) ): T-Value \(=1.87\) P-Value \(=0.066 \quad \mathrm{DF}=54\)

We see } in the AllCountries dataset that the percent of the population living in rural areas is 8.0 in Argentina and 34.4 in Bolivia. Suppose we take random samples of size 200 from each country, and compute the difference in sample proportions \(\hat{p}_{A}-\hat{p}_{B},\) where \(\hat{p}_{A}\) represents the sample proportion living in rural areas in Argentina and \(\hat{p}_{B}\) represents the proportion of the sample that lives in rural areas in Bolivia. (a) Find the mean and standard deviation of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) If the sample sizes are large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. (c) Using the graph drawn in part (b), are we likely to see a difference in sample proportions as large in magnitude as -0.4 ? As large as \(-0.3 ?\) Explain.
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