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Chapter 9: Problem 40

Tomatoes Use the data from exercise \(9.36\). a. Using the four-step procedure with a two-sided alternative hypothesis, should you be able to reject the hypothesis that the population mean is 5 pounds using a significance level of \(0.05\) ? Why or why not? The confidence interval is reported here: \(\mathrm{I}\) am \(95 \%\) confident the population mean is between \(4.9\) and \(5.3\) pounds. b. Now test the hypothesis that the population mean is not 5 pounds using the four-step procedure. Use a significance level of \(0.05\) and number your steps.

Short Answer

Expert verified
a. No, we don't reject the null hypothesis that the population mean is 5 pounds because the confidence interval includes 5. b. Following the four-step procedure, we similarly do not reject the null hypothesis that the population mean is not 5 pounds.

Step by step solution

01

State the Hypotheses

For both parts a and b, the null hypothesis \(H_0\) is that the population mean is 5 pounds, i.e., \(\mu = 5\). The alternative hypothesis \(H_a\) is that the population mean is not 5 pounds, i.e., \(\mu \neq 5\).
02

Formulate the Analysis Plan

From the given information, a confidence interval of the population mean is between 4.9 and 5.3 pounds. The significance level is set to 0.05.
03

Analyze Sample Data

Since we have the range of confidence interval, we know that the mean is within the range of 4.9 and 5.3 pounds. If the value 5 (from null hypothesis) falls within this range, then we do not reject the null hypothesis.
04

Interpret the Result

Because 5 does lay within the range of 4.9 to 5.3 pounds (our confidence interval). So, for part a, we do not reject the null hypothesis that the population mean is 5 pounds. Part b proceeds in a similar method; it's just a reiteration of the previous procedure but needs to number the steps.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis serves as a statement that there is no effect or no difference in the population being studied. In this exercise, the null hypothesis \(H_0\) is that the population mean weight of the tomatoes is 5 pounds. This is expressed mathematically as \(\mu = 5\). The intention of hypothesis testing is to determine whether there is sufficient evidence to reject this statement.However, the null hypothesis typically assumes that any observed difference in sample data is purely due to randomness.
  • For a hypothesis test, failure to reject \(H_0\) implies there is not enough evidence against it.
  • Rejecting \(H_0\) suggests support for the alternative hypothesis, \(H_a\).
Understanding the null hypothesis is fundamental, as it guides the analysis and interpretation of data results.
Confidence Interval
A confidence interval provides a range of values for the estimated parameter, highlighting the uncertainty associated with the estimate. In this exercise, the confidence interval for the population mean of tomato weights is between 4.9 and 5.3 pounds. A confidence interval is typically expressed with a percentage, known as the confidence level, which quantifies our certainty that the true mean lies within this range. A 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each, we expect about 95 of them to contain the true population mean.
  • The width of a confidence interval is influenced by the sample size and variability of the data.
  • A narrow interval suggests a more precise estimate of the population parameter.
  • In this case, because 5 pounds falls within the range, there isn't enough evidence to reject the null hypothesis.
Population Mean
The population mean is a critical parameter in statistics, representing the average of all individual measurements in a complete population. For this experiment, the population mean \(\mu\) is the average weight of the entire population of tomatoes.Usually, researchers use sample data to make inferences about the population mean, due to the impracticality of measuring every individual within a population.Key elements about population mean in hypothesis testing include:
  • It serves as a benchmark for assessing evidence against the null hypothesis.
  • Fluctuations around the sample mean provide insights into the confidence interval.
  • The accuracy of the estimated population mean largely depends on the sample size鈥攍arger samples typically lead to a more reliable estimate.
In this exercise, the hypothesis testing begins with the premise that \(\mu = 5\) pounds, and the confidence interval helps assess the credibility of this claim.
Significance Level
The significance level, often denoted as \(\alpha\), is a threshold used in hypothesis testing to decide whether to reject the null hypothesis. It represents the probability of making a Type I error鈥攔ejecting the null hypothesis when it is actually true.In this exercise, the significance level is set at 0.05. This means that there is a 5% chance of incorrectly rejecting the null hypothesis.
  • The lower the significance level, the stricter the criterion for rejecting \(H_0\).
  • A common choice for \(\alpha\) is 0.05; however, more stringent fields may choose 0.01 or even 0.001.
  • Interpreting test results depends considerably on the chosen significance level.
By choosing an appropriate \(\alpha\), researchers balance the risk of Type I errors with practical considerations in hypothesis testing.

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Most popular questions from this chapter

Medical School Acceptance Rates The acceptance rate for a random sample of 15 medical schools in the United States was taken. The mean acceptance rate for this sample was \(5.77\) with a standard error of \(0.56 .\) Assume the distribution of acceptance rates is Normal. (Source: Accepted.com) a. Decide whether cach of the following statements is worded correctly for the confidence interval. Fill in the blanks for the correctly worded one(s). Explain the error for the ones that are incorrectly worded. i. We are \(95 \%\) confident that the sample mean is between and ii. We are \(95 \%\) confident that the population mean is between \(\quad\) and iii. There is a \(95 \%\) probability that the population mean is between and b. Based on your confidence interval, would you believe the average acceptance rate for medical schools is \(6.5\) ? Explain.

Exam Scores The distribution of the scores on a certain exam is \(N(100,10)\) which means that the exam scores are Normally distributed with a mean of 100 and a standard deviation of \(10 .\) a. Sketch or use technology to create the curve and label on the \(x\) -axis the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score is between 90 and 110 . Shade the region under the Normal curve whose area corresponds to this probability.

Student Ages The mean age of all 2550 students at a small college is \(22.8\) years with a standard deviation is \(3.2\) years, and the distribution is right- skewed. A random sample of 4 students' ages is obtained, and the mean is \(23.2\) with a standard deviation of \(2.4\) years. a. \(\mu=? \quad \sigma=? \quad \bar{x}=? \quad s=?\) b. Is \(\mu\) a parameter or a statistic? \(c\). Are the conditions for using the CLT fulfilled? What would be the shape of the approximate sampling distribution of many means, cach from a sample of 4 students? Would the shape be right-skewed, Normal, or left-skewed?

Pulse and Gender: Cl Using data from NHANES, we looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. NHANES data are random and independent. Minitab output follows.

Movie Ticket Prices According to Deadline.com, the average price for a movie ticket in 2018 was \(\$ 8.97 .\) A random sample of movie prices in the San Francisco Bay Area 25 movie ticket prices had a sample mean of \(\$ 12.27\) with a standard deviation of \(\$ 3.36\). a. Do we have evidence that the price of a movie ticket in the San Francisco Bay Area is different from the national average? Use a significance level of \(0.05\). b. Construct a \(95 \%\) confidence interval for the price of a movic ticket in the San Francisco Bay Area. How does your confidence interval support your conclusion in part a?
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