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Chapter 9: Problem 13

Attitudes In the study of older students鈥� attitudes from Exercise 3, the sample mean SSHA score was 125.7 and the sample standard deviation was 29.8. A significance test yields a P-value of 0.0101. (a) Interpret the P-value in context. (b) What conclusion would you make if \(\alpha=0.05\) ? If \(\alpha=0.01\) Justify your answer.

Short Answer

Expert verified
Reject null hypothesis at \(\alpha=0.05\), do not reject at \(\alpha=0.01\).

Step by step solution

01

Understand the P-value

The P-value is 0.0101, which indicates the probability of observing a sample mean as extreme as 125.7, given that the null hypothesis is true. In context, a P-value of 0.0101 means there is approximately a 1.01% chance that the data we observed (or something more extreme) would occur if the true population mean were actually the same as the one under our null hypothesis.
02

Setting Alpha Levels for Decision-Making

We have two alpha levels to consider: \(\alpha = 0.05\) and \(\alpha = 0.01\). These values are thresholds for determining whether the P-value indicates a statistically significant result. Comparing the P-value to these thresholds will help us determine if we should reject the null hypothesis.
03

Compare P-value with Alpha = 0.05

With \(\alpha = 0.05\), the P-value is 0.0101, which is less than 0.05. This means that the result is statistically significant at the 5% level, and we have enough evidence to reject the null hypothesis.
04

Compare P-value with Alpha = 0.01

With \(\alpha = 0.01\), the P-value is 0.0101, which is greater than 0.01. This means that the result is not statistically significant at the 1% level, and we do not have enough evidence to reject the null hypothesis.
05

Conclusion Based on Each Alpha Level

For \(\alpha = 0.05\), we reject the null hypothesis and conclude that there is significant evidence at the 5% level. For \(\alpha = 0.01\), we do not reject the null hypothesis, indicating there isn't enough evidence to support the alternative hypothesis at the 1% level.

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Key Concepts

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

Statistical Significance
The idea of statistical significance is central to hypothesis testing. When conducting a test, we want to see if our sample data gives us results that make us question the status quo, known as the null hypothesis. Statistical significance helps us determine whether the observed data is due to a real effect or just occurred by random chance.
For example, in our exercise, the P-value was 0.0101, which we compared against certain alpha levels like 0.05 and 0.01. The P-value indicated whether the sample mean of 125.7 was significant enough to reject the null hypothesis.
  • When a result is statistically significant, it does not mean it's important or practical, just that it is unlikely to have occurred under the null hypothesis.
  • How "unlikely" is determined by our chosen threshold, the alpha level.
Null Hypothesis
Understanding the null hypothesis is key to making sense of hypothesis tests. The null hypothesis, often noted as \( H_0 \), is the assumption that there is no effect or no difference, and any observed effect is due to sampling or experimental error. In the context of the given exercise, the null hypothesis would state that the true population mean is a specific value we believe to be true unless proven otherwise by the test results.
  • It's important because it sets a baseline for comparison.
  • The test aims to reject \( H_0 \), which implies evidence for an alternative hypothesis, \( H_a \).
  • A non-rejection of \( H_0 \) doesn't prove it true; it simply means we lack strong evidence against it.
Alpha Level
The alpha level, symbolized as \( \alpha \), serves as a threshold in hypothesis testing. It defines the maximum probability we allow for incorrectly rejecting the null hypothesis when it's true, known as a Type I error. Commonly used alpha levels are 0.05 and 0.01.
In our exercise, two alpha levels were used to compare against the P-value of 0.0101:
  • \(\alpha = 0.05\): The result was statistically significant, prompting a rejection of the null hypothesis.
  • \(\alpha = 0.01\): The result was not statistically significant, indicating insufficient evidence to reject the null hypothesis.
Thus, choosing an alpha level is crucial because it affects the stringency of our test.
  • A smaller alpha level (e.g., 0.01) means less risk of a Type I error but requires stronger evidence to reject \( H_0 \).
  • A larger alpha level (e.g., 0.05) allows more room for Type I error but can be too lenient for certain studies.
Sample Mean
The sample mean is the average of all the data points in a sample and is a crucial statistic in hypothesis testing. It serves as an estimate of the population mean when analyzing data. In our exercise, a sample mean of 125.7 was identified. This value became the central focus in evaluating whether the sample provided evidence against the null hypothesis.
  • Calculated by summing all data points and dividing by the number of points: \( \bar{x} = \frac{\sum_{i=1}^{n}x_i}{n} \).
  • It reflects the central tendency of the sample data.
  • If different from the hypothesized population mean, it suggests a possible effect.
The sample mean can be influenced by outliers, so always consider data distributions when interpreting it.

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Most popular questions from this chapter

Power A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer鈥檚 disease will experience nausea. To test this claim, a significance test is carried out of $$ \begin{array}{l}{H_{0} : p=0.10} \\ {H_{a} : p<0.10}\end{array} $$ You learn that the power of this test at the 5\(\%\) significance level against the alternative \(p=0.08\) is 0.64 (a) Explain in simple language what "power \(=0.64^{\prime \prime}\) means in this setting. (b) You could get higher power against the same alternative with the same \(\alpha\) by changing the number of measurements you make. Should you make more measurements or fewer to increase power? Explain. (c) If you decide to use \(\alpha=0.01\) in place of \(\alpha=\) \(0.05,\) with no other changes in the test, will the power increase or decrease? Justify your answer. (d) If you shift your interest to the alternative \(p=\) 0.07 with no other changes, will the power increase or decrease? Justify your answer.

Are boys more likely? We hear that newborn babies are more likely to be boys than girls. Is this true? A random sample of 25,468 firstborn children included 13,173 boys.13 Boys do make up more than half of the sample, but of course we don鈥檛 expect a perfect 50-50 split in a random sample. (a) To what population can the results of this study be generalized: all children or all firstborn children? Justify your answer. (b) Do these data give convincing evidence that boys are more common than girls in the population? Carry out a significance test to help answer this question.

Error probabilities You read that a statistical test at the \(\alpha=0.01\) level has probability 0.14 of making a Type II error when a specific alternative is true. What is the power of the test against this alternative?

Ages of presidents Joe is writing a report on the backgrounds of American presidents. He looks up the ages of all the presidents when they entered office. Because Joe took a statistics course, he uses these numbers to perform a significance test about the mean age of all U.S. presidents. Explain why this makes no sense.

Filling cola bottles Bottles of a popular cola are supposed to contain 300 milliliters (ml) of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. From experience, the distribution of the contents is approximately Normal. An inspector measures the contents of six randomly selected bottles from a single day鈥檚 production. The results are 299.4\(\quad 297.7 \quad 301.0 \quad 298.9 \quad 300.2 \quad 297.0\) Do these data provide convincing evidence that the mean amount of cola in all the bottles filled that day differs from the target value of 300 \(\mathrm{ml}\) ? Carry out an appropriate test to support your answer.
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