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 mg/100 ml). 95 90 84 107 97 108 82 91 The sample mean is x ≈ 94.3. 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 ? = 12.5. The mean glucose level for horses should be ? = 85 mg/100 ml.† Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use ? = 0.05. State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: ? = 85; H1: ? ≠ 85; two-tailed H0: ? = 85; H1: ? < 85; left-tailed H0: ? > 85; H1: ? = 85; right-tailed H0: ? = 85; H1: ? > 85; right-tailed What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. A) The standard normal, since we assume that x has a normal distribution with unknown ?. B) The Student's t, since we assume that x has a normal distribution with known ?. C) The Student's t, since n is large with unknown ?. D) The standard normal, since we assume that x has a normal distribution with known ?. Compute the z value of the sample test statistic. (Round your answer to two decimal places.) Find (or estimate) the P-value. (Round your answer to four decimal places.) 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 ?? A) At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. B) At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. C) At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. D) At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
95 | 90 | 84 | 107 | 97 | 108 | 82 | 91 |
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