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Chapter 8: Problem 17

Pasteurized milk may not have a standardized plate count (SPC) above 20,000 colony-forming bacteria per milliliter (cfu/ml). The mean SPC for five samples was \(21,500 \mathrm{cfu} / \mathrm{ml}\) with sample standard deviation \(750 \mathrm{cfu} / \mathrm{m} 1\). Test the null hypothesis that the mean \(\mathrm{SPC}\) for this milk is 20,000 versus the alternative that it is greater than 20,000 , at the \(10 \%\) level of significance. Assume that the SPC follows a normal distribution.

Short Answer

Expert verified
Reject the null hypothesis; SPC is likely greater than 20,000 cfu/ml at 10% significance level.

Step by step solution

01

State the Hypotheses

In hypothesis testing, we start by stating the null and alternative hypotheses. Here, our null hypothesis is that the mean SPC is 20,000, and the alternative hypothesis is that the mean SPC is greater than 20,000.\[H_0: \mu = 20000 \H_a: \mu > 20000\]
02

Determine the Test Statistic

Since we know the sample mean, sample standard deviation, and sample size, we use the t-statistic formula to find the test statistic:\[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}\]Given:- Sample mean, \(\bar{x} = 21500\)- Hypothesized mean, \(\mu_0 = 20000\)- Sample standard deviation, \(s = 750\)- Sample size, \(n = 5\)Substituting the values:\[t = \frac{21500 - 20000}{750/\sqrt{5}} = \frac{1500}{335.41} \approx 4.47\]
03

Determine the Critical Value

We now determine the critical value for a one-tailed test at the 10% significance level (\(\alpha = 0.10\)). Typically, we use a t-distribution with \(n-1 = 4\) degrees of freedom.Looking up the t-table, the critical t-value for \(df = 4\) and \(\alpha = 0.10\) is approximately 1.533.
04

Make the Decision

Compare the calculated t-statistic to the critical value. If the t-statistic is greater than the critical value, we reject the null hypothesis. Here: - Calculated t-statistic = 4.47 - Critical t-value = 1.533 Since 4.47 > 1.533, we reject the null hypothesis.
05

Conclusion

Based on our test, we reject the null hypothesis that the mean SPC is 20,000. This suggests, at the 10% significance level, that the mean SPC is indeed greater than 20,000.

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Key Concepts

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

T-Distribution
The t-distribution is a critical tool in hypothesis testing when dealing with small sample sizes. It's a bit like its cousin, the normal distribution, but it spreads out a bit more and has fatter tails, which accommodates the additional uncertainty from estimating the population standard deviation.
For samples around 30 or less, the t-distribution is ideal. As our sample size increases, the t-distribution approaches the normal distribution. This adaptability makes the t-distribution vital when handling smaller data sets.
  • It is symmetric, just like the normal distribution.
  • The mean of the distribution is zero, but the tails are thicker, making extreme values more probable.
  • As the degrees of freedom increase, the t-distribution becomes closely aligned with the normal distribution.
Critical Value
When conducting hypothesis tests, the critical value helps determine whether to reject the null hypothesis. It acts like a threshold or a cut-off point. In a t-test, this value is drawn from the t-distribution table corresponding to the degrees of freedom and the chosen significance level.
In our context, with a sample size of 5, the degrees of freedom is calculated as 4. We choose the 10% significance level for a one-tailed test. From the t-distribution table, the critical value at this point is 1.533.
If your test statistic exceeds this critical value, you have grounds to reject the null hypothesis. Here, the test statistic was 4.47, clearly larger than 1.533, leading to the rejection of the null hypothesis.
Null Hypothesis
The null hypothesis, often symbolized as \(H_0\), represents a statement of no effect or no difference. It's the hypothesis that the test aims to either disprove or fail to disprove. Establishing this is a crucial first step in hypothesis testing.
In the given exercise, the null hypothesis \(H_0: \mu = 20000\) asserts that the mean SPC is 20,000 cfu/ml. This provides a standard or benchmark against which the sample mean is tested.
If the evidence from the data consistently contradicts this hypothesis, such as a significantly higher sample mean, the null hypothesis is rejected in favor of the alternative.
Alternative Hypothesis
The alternative hypothesis, denoted by \(H_a\), is what you hope to support with your data evidence. This hypothesis represents a new finding or effect that you're setting up to prove. In hypothesis testing, rejecting the null hypothesis lends support to the alternative hypothesis.
For our milk SPC example, \(H_a: \mu > 20000\) suggests that the mean SPC is greater than 20,000 cfu/ml. This was the hypothesis we ended up supporting, as the test statistic indicated a higher mean.
  • It is directional, suggesting some difference greater than, less than, or not equal to the null hypothesis value.
  • In the one-tailed test, as seen here, it explicitly specifies a direction (e.g., greater than).
  • The alternative hypothesis is only supported via rejection of the null hypothesis.

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Most popular questions from this chapter

An economist wishes to determine whether people are driving less than in the past. In one region of the country the number of miles driven per household per year in the past was 18.59 thousand miles. A sample of 15 households produced a sample mean of 16.23 thousand miles for the last year, with sample standard deviation 4.06 thousand miles. Assuming a normal distribution of household driving distances per year, perform the relevant test at the \(5 \%\) level of significance.

An insurance company states that it settles \(85 \%\) of all life insurance claims within 30 days. A consumer group asks the state insurance commission to investigate. In a sample of 250 life insurance claims, 203 were settled within 30 days. a. Test whether the true proportion of all life insurance claims made to this company that are settled within 30 days is less than \(85 \%,\) at the \(5 \%\) level of significance. b. Compute the observed significance of the test.

The government of an impoverished country reports the mean age at death among those who have survived to adulthood as 66.2 years. A relief agency examines 30 randomly selected deaths and obtains a mean of 62.3 years with standard deviation 8.1 years. Test whether the agency's data support the alternative hypothesis, at the \(1 \%\) level of significance, that the population mean is less than 66.2 .

Large Data Set 1 lists the SAT scores of 1,000 students. a. Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean \(\mu\). b. Regard the first 50 students in the data set as a random sample drawn from the population of part (a) and use it to test the hypothesis that the population mean exceeds \(1,510,\) at the \(10 \%\) level of significance. (The null hypothesis is that \(\mu=1510 .\) ) (a) you c. Is your conclusion in part (b) in agreement with the true state of nature (which by part know), or is your decision in error? If your decision is in error, is it a Type I error or a Type II error?

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number \(\mu_{0}\) and write \(H_{0} \cdot \mu=\mu_{0}\) and the appropriate analogous expression for \(H_{a}\).) a. The average July temperature in a region historically has been \(74.5^{\circ} \mathrm{F}\). Perhaps it is higher now. b. The average weight of a female airline passenger with luggage was 145 pounds ten years ago. The FAA believes it to be higher now. c. The average stipend for doctoral students in a particular discipline at a state university is \(\$ 14,756\). The department chairman believes that the national average is higher. d. The average room rate in hotels in a certain region is \(\$ 82.53 .\) A travel agent believes that the average in a particular resort area is different. e. The average farm size in a predominately rural state was 69.4 acres. The secretary of agriculture of that state asserts that it is less today.
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