91Ó°ÊÓ

The calibration of a scale is to be checked by weighing a \(10-\mathrm{kg}\) test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with \(\sigma=.200 \mathrm{~kg}\). Let \(\mu\) denote the true average weight reading on the scale. a. What hypotheses should be tested? b. Suppose the scale is to be recalibrated if either \(\bar{x} \geq 10.1032\) or \(\bar{x} \leq 9.8968\). What is the probability that recalibration is carried out when it is actually unnecessary? c. What is the probability that recalibration is judged unnecessary when in fact \(\mu=10.1\) ? When \(\mu=9.8\) ? d. Let \(z=(\bar{x}-10) /(\sigma / \sqrt{n})\). For what value \(c\) is the rejection region of part (b) equivalent to the "two-tailed" region of either \(z \geq c\) or \(z \leq-c\) ? e. If the sample size were only 10 rather than 25 , how should the procedure of part (d) be altered so that \(\alpha=.05\) ? f. Using the test of part (e), what would you conclude from the following sample data? $$ \begin{array}{rrrrr} 9.981 & 10.006 & 9.857 & 10.107 & 9.888 \\ 9.728 & 10.439 & 10.214 & 10.190 & 9.793 \end{array} $$ g. Reexpress the test procedure of part (b) in terms of the standardized test statistic \(Z=(\bar{X}-10) /(\sigma / \sqrt{n})\).

Step by step solution

01

Define Hypotheses

We want to test whether the scale is calibrated properly around the true weight of 10 kg. The null hypothesis (H0) should state that the scale is correctly calibrated, i.e., \( \mu = 10 \). The alternative hypothesis (H1) is that the scale is not correctly calibrated, i.e., \( \mu eq 10 \).

02

Probability of Unnecessary Recalibration

We are given that recalibration occurs if \( \bar{x} \geq 10.1032 \) or \( \bar{x} \leq 9.8968 \). These are critical values set for the specified distribution. For the true mean \( \mu = 10 \), use the formula \( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \) to find the critical region in terms of \( z \). Given \( \sigma = 0.200 \) and \( n = 25 \), we perform a normal distribution calculation to find the probability. This is a two-tailed test.

03

Probability of Missed Recalibration

For \( \mu = 10.1 \) and \( \mu = 9.8 \), calculate the probability that \( \bar{x} \) will be between 9.8968 and 10.1032. This is the chance of not recalibrating when it is needed. Use the distribution formula \( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \) for each \( \mu \) case and find the probability using the cumulative distribution function (CDF).

04

Determine Two-tailed Test Constant

We know \( z = \frac{\bar{x} - 10}{\sigma/\sqrt{n}} \) and need to find the constant \( c \) such that the region is \( z \geq c \) or \( z \leq -c \). Solving it for the set intervals gives that the values of \( c \) ends up being \( \pm \frac{0.1032}{0.200/\sqrt{25}} \).

05

Adjust Procedure for Smaller Sample

If the sample size is \( n = 10 \), adjust \( \alpha = 0.05 \). With a smaller sample size, we would still calculate critical \( z \) values related to this new configuration but with reduced degrees of freedom, using a t-distribution instead.

06

Test Sample Data

For the given data, calculate \( \bar{x} \). Then find the test statistic \( Z \) using \( z = \frac{\bar{x} - 10}{0.200/\sqrt{10}} \). Compare this to benchmark region to conclude whether recalibration is necessary for the sample data.

07

Standardized Test Statistic Expression

Restate the criteria for part (b) in a standardized form using \( Z = \frac{\bar{X} - 10}{0.200 / \sqrt{25}} \). The recalibration rule becomes \( Z \geq c \) or \( Z \leq -c \), where \( c \) corresponds to calculated critical value from \( \bar{x} \).

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Key Concepts

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

Normal Distribution

When we talk about the normal distribution in statistics, we're referring to a specific type of continuous probability distribution. It's often called the bell curve due to its shape. Here are some key points about the normal distribution:

• **Characteristics**: Symmetrical around the mean, with the mean, median, and mode all being equal.
• **Standard Deviation**: Determines the spread of the distribution. Most data points lie close to the mean in a normal distribution.
• **Probability Density Function**: It's defined mathematically by the function \( f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{(x-\mu)^2/2\sigma^2} \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

In the context of our exercise, each weighing result is expected to follow a normal distribution with a specified mean and standard deviation. This helps in assessing how likely the observed results reflect a properly calibrated scale.

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that there is no effect or no difference, and it's what you aim to test. The null hypothesis is usually denoted as \( H_0 \).

• **Definition**: In many cases, the null hypothesis represents a default or 'no effect' scenario.
• **Purpose**: It serves as the starting point for statistical testing. We assume it is true until we find evidence to suggest otherwise.
• **Example**: In our exercise, the null hypothesis is \( H_0: \mu = 10 \), stating that the scale is correctly calibrated with an average reading of 10 kg.

This hypothesis is tested to judge whether any deviation observed in the sample mean is due to random sampling error or indicates that the scale is indeed off-calibration.

Alternative Hypothesis

The alternative hypothesis is the stark contrast to the null hypothesis. It proposes that there is indeed a significant effect or difference.

• **Notation**: Represented as \( H_1 \) or \( H_a \).
• **Objective**: To propose a different scenario if the null hypothesis is proven false.
• **Example**: For our exercise, the alternative hypothesis is \( H_1: \mu eq 10 \). It suggests that the scale may not be properly calibrated and that the average weight reading differs from 10 kg.
• **Outcome**: Proving the alternative hypothesis often leads to action, such as recalibrating the scale.

This is a critical part of hypothesis testing, as action is typically based on whether the alternative hypothesis is accepted.

Two-Tailed Test

A two-tailed test checks for any statistically significant deviation from the null hypothesis in both directions. This means testing for both an increase and a decrease in the parameter of interest.

• **Definition**: Examines changes in two directions, making it more sensitive to finding any deviation.
• **Use**: Appropriate when you care about any deviation from the specified parameter, not just an increase or decrease.
• **Application**: In the scale exercise, a two-tailed test is used to determine if the scale reads either significantly above or below 10 kg, indicating a calibration issue.
• **Implication**: Essentially doubles the regions of rejection in the probability distribution, making test processes like recalibration legitimate for both higher and lower deviations from 10 kg.

This testing method provides a comprehensive approach to ensure that any out-of-calibration reading is effectively captured, regardless of its direction.

Critical Region

The critical region in hypothesis testing is the range of values that leads to the rejection of the null hypothesis.

• **Definition**: It's a portion of the distribution where, if the test statistic falls, we reject the null hypothesis.
• **Critical Values**: These are the boundaries of the critical region, set based on significance level \( \alpha \).
• **Role in Calculation**: Determining this region helps decide whether the observed data leads to rejecting \( H_0 \).
• **Example**: In the exercise, critical regions correspond to weight readings where the scale is considered out of calibration, below 9.8968 or above 10.1032.

Identifying and applying the critical region effectively aids in precisely making statistical decisions, reflecting whether a hypothesis should stand or be rejected.