91Ó°ÊÓ

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the \(z\) value of the sample test statistic. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in \(\mathrm{mg} / 100 \mathrm{ml}\) ). The sample mean is \(\bar{x} \approx 93.8 .\) Let \(x\) be a random variable representing glucose readings taken from Gentle Ben. We may assume that \(x\) has a normal distribution, and we know from past experience that \(\sigma=12.5 .\) The mean glucose level for horses should be \(\mu=85 \mathrm{mg} / 100 \mathrm{ml}\) (Reference: Merck Veterinary Manual). Do these data indicate that Gentle Ben has an overall average glucose level higher than \(85 ?\) Use \(\alpha=0.05\)

Step by step solution

01

Identify the Level of Significance and Formulate Hypotheses

The level of significance is given by \( \alpha = 0.05 \). We need to formulate the null and alternative hypotheses. The null hypothesis is \( H_0: \mu = 85 \), while the alternative hypothesis is \( H_1: \mu > 85 \), as we want to know if the glucose level is higher. Since we are checking if the mean glucose level is higher than 85, it is a right-tailed test.

02

Determine the Sampling Distribution and Calculate Test Statistic

Since the sample mean is used and we know the population standard deviation \( \sigma = 12.5 \), we will use the normal distribution. The test statistic \( z \) is calculated using the formula: \[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \] where \( \bar{x} = 93.8 \), \( \mu = 85 \), and \( n \) is the sample size (not given directly, but typically a large \( n \) if using normal distribution). Assuming \( n = 8 \) as indicated by past readings, the calculation becomes: \[ z = \frac{93.8 - 85}{12.5 / \sqrt{8}} \approx 1.983 \].

03

Estimate or Find the P-value and Visualize

The \( P \)-value corresponds to the right tail beyond the calculated \( z \)-value in the standard normal distribution. Using \( z \approx 1.983 \), we find \( P(z > 1.983) \) using statistical tables or software, which gives \( P \approx 0.0239 \). This area represents the likelihood of observing a sample mean of 93.8 or more if the true mean is 85. This would be visualized as a shaded area to the right of \( z = 1.983 \) on a standard normal distribution curve.

04

Decision Rule in Hypothesis Testing

Since the calculated \( P \)-value (0.0239) is less than the level of significance \( \alpha = 0.05 \), we reject the null hypothesis \( H_0 \). Therefore, the data are statistically significant at the 0.05 level, indicating that Gentle Ben's average glucose level is likely higher than 85.

05

Interpret Results in Context

Based on the statistical test, we conclude that with 95% confidence, the glucose level readings suggest that Gentle Ben has an overall average glucose level higher than the typical 85 mg/100 ml, possibly indicating a medical concern or deviation from the expected norm for horses.

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Key Concepts

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

Level of Significance

When conducting a hypothesis test, setting a level of significance is essential. It is denoted by \( \alpha \) and represents the threshold of risk we are willing to accept for making a type I error — that is, rejecting the null hypothesis when it's actually true. A common value for \( \alpha \) is 0.05, which implies a 5% chance of making this error. In our example with Gentle Ben, we set \( \alpha = 0.05 \), meaning there is a 5% risk that we might say his glucose levels are higher than normal when they are not.

It's crucial to clarify your hypotheses:

• The null hypothesis (\( H_0 \)) usually states that there is no effect or difference, in this case, that Gentle Ben’s mean glucose level \( \mu \) equals 85.
• The alternative hypothesis (\( H_1 \)) proposes what you suspect might be true, which here is that \( \mu > 85 \).
  
  This setup facilitates deciding whether any statistical observation is sufficiently surprising to suggest an alternative explanation.

Z-Test

A z-test is a statistical test used to determine if there is a significant difference between sample and population means, assuming a normal distribution and known standard deviation. For Gentle Ben, the glucose readings follow this normal distribution, making the z-test appropriate.

The formula used is:
\[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \] where:

• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean
• \( \sigma \) is the population standard deviation
• \( n \) is the sample size

This calculation helps us determine how far our sample mean (93.8) deviates from the assumed population mean (85) in terms of standard errors. In Gentle Ben's case, a z-value of approximately 1.983 means his glucose readings are nearly 2 standard errors above the mean for horses. This z-test thus quantifies our observed difference.

P-Value

The p-value helps us assess the strength of our results. It indicates the probability that the observed or more extreme result would occur purely by chance if the null hypothesis were true.

For a right-tailed test like ours, we look for \( P(z > 1.983) \). Using statistical software or z-tables, we find \( P \approx 0.0239 \), indicating a 2.39% chance of getting such a result by random chance if the null hypothesis is true.

When compared to our \( \alpha \) of 0.05, the p-value is smaller, meaning the observational data is too rare under the null hypothesis, thus leading us to reject \( H_0 \). The smaller the p-value, the stronger the evidence against the null hypothesis. However, it's important to remember that a low p-value doesn't measure the magnitude or size of an effect.

Sampling Distribution

In hypothesis testing, the sampling distribution of a statistic, like the mean, shows how a statistic would behave if we repeatedly drew random samples size \( n \) from a population. Here, because we know the population standard deviation (12.5), the sampling distribution is normal.

The Central Limit Theorem justifies us using this normal distribution given enough sample size, which approximates the distribution of sample means, regardless of the original population's distribution if \( n \) is large.

In our example, the z-distribution of Gentle Ben’s glucose readings is built on these assumptions. As the z-value progresses along the x-axis, we assess the likelihood of observing our sample mean compared to the population mean, indicating what's likely or just purely by chance.

Right-Tailed Test

A right-tailed test is used when we expect the parameter in question (here, glucose level \( \mu \)) to be greater than a specified value. For Gentle Ben's high glucose level check, a right-tailed test assesses if the sample mean is significantly greater than known level (85).

This type of tailing looks at values that are greater, meaning we check whether the sample provides enough evidence to suggest the parameter exceeds the threshold.

Visualizing this, it means checking the area under the normal curve to the right of our calculated z, visualized as where improbable, extreme, or high values reside. If this area (p-value) is smaller than \( \alpha \), the null hypothesis is rejected, reinforcing the presence of a statistically significant higher mean.