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Chapter 9: Problem 34

Briefly explain the conditions that must hold true to use the \(t\) distribution to make a test of hypothesis about the population mean.

Short Answer

Expert verified
To use the t-distribution for hypothesis testing about the population mean, the following conditions must hold true: the sample size is less than 30, the population is normally or approximately normally distributed, the population standard deviation is unknown, the data sample is random, and the data points are independent of each other.

Step by step solution

01

Condition 1: Sample Size

The size of the sample should be less than 30 (it is a small sample). This condition arises because the t-distribution is generally used when dealing with small sample sizes.
02

Condition 2: Population Distribution

The population from which the samples are taken should follow a normal distribution or be approximately normally distributed. This condition ensures that the sample mean would also follow a normal distribution, a fundamental requirement for the test.
03

Condition 3: Unknown population standard deviation

Use the t-distribution when the standard deviation of the population (\(\sigma\)) is unknown and it is approximated by the sample's standard deviation (s).
04

Condition 4: Random Sampling

The selection of the data must be done through a method of random sampling, meaning every member of the population has an equal chance of getting selected. Random sampling ensures the representation of the sample reflects the population to the maximum possible extent.
05

Condition 5: Independence

Data points in the sample have to be independent of each other. This condition is critical because the influence of one data point on another can distort the results of the hypothesis test.

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Most popular questions from this chapter

In a Gallup poll conducted July \(7-10,2014,45 \%\) of Americans said that they actively try to include organic foods into their diets (www.gallup.com). In a recent sample of 2100 Americans, 1071 said that they actively try to include organic foods into their diets. Is there significant evidence at a \(1 \%\) significance level to conclude that the current percentage of all Americans who will say that they actively try to include organic foods into their diets is different from \(45 \%\) ? Use both the \(p\) -value and the critical-value approaches.

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=60 \text { versus } \quad H_{1}: \mu>60 $$ Suppose you perform this test at \(\alpha=.01\) and fail to reject the null hypothesis. Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is "statistically significant" or would you state that this difference is "statistically not significant?" Explain.

What are the five steps of a test of hypothesis using the critical value approach? Explain briefly,

At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton but has an approximate normal distribution. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the milk in all such cartons is always equal to 15 ounce. The quality control inspector at this company takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and tests the null hypothesis, \(\mu=32\) ounces, against the altemative hypothesis, \(\mu \neq 32\) ounces. If the null hypothesis is rejected, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.93\) ounces. a. Calculate the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, will the quality control inspector decide to stop the machine and adjust it if she chooses the maximum probability of a Type I error to be .01? What if the maximum probability of a Type I error is 05 ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\). Does the machine need to be adjusted? What if \(a=.05 ?\)

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=12, H_{1}: \mu<12\) b. \(H_{0}: \mu \leq 85, H_{1}: \mu>85\) c. \(H_{0}: \mu=33, H_{1}: \mu \neq 33\) Show the rejection and nonrejection regions for each of these cases by drawing a sampling distribution curve for the sample mean, assuming that it is normally distributed.
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