Hypothesis Test Example of Calculating Probability

Hypothesis Test Example of Calculating Probability An important part of inferential statistics is hypothesis testing. As with learning anything related to mathematics, it is helpful to work through several examples. The following examines an example of a hypothesis test, and calculates the probability of type I and type II errors. We will assume that the simple conditions hold. More specifically we will assume that we have a simple random sample from a population that is either normally distributed or has a large enough sample size that we can apply the central limit theorem. We will also assume that we know the population standard deviation. Statement of the Problem A bag of potato chips is packaged by weight. A total of nine bags are purchased, weighed and the mean weight of these nine bags is 10.5 ounces. Suppose that the standard deviation of the population of all such bags of chips is 0.6 ounces. The stated weight on all packages is 11 ounces. Set a level of significance at 0.01. Question 1 Does the sample support the hypothesis that true population mean is less than 11 ounces? We have a lower tailed test. This is seen by the statement of our null and alternative hypotheses: H0 : ÃŽ ¼11.Ha : ÃŽ ¼ 11. The test statistic is calculated by the formula z (x-bar - ÃŽ ¼0)/(ÏÆ'/√n) (10.5 - 11)/(0.6/√ 9) -0.5/0.2 -2.5. We now need to determine how likely this value of z is due to chance alone. By using a table of z-scores we see that the probability that z is less than or equal to -2.5 is 0.0062. Since this p-value is less than the significance level, we reject the null hypothesis and accept the alternative hypothesis. The mean weight of all bags of chips is less than 11 ounces. Question 2 What is the probability of a type I error? A type I error occurs when we reject a null hypothesis that is true. The probability of such an error is equal to the significance level. In this case, we have a level of significance equal to 0.01, thus this is the probability of a type I error. Question 3 If the population mean is actually 10.75 ounces, what is the probability of a Type II error? We begin by reformulating our decision rule in terms of the sample mean. For a significance level of 0.01, we reject the null hypothesis when z -2.33. By plugging this value into the formula for the test statistics, we reject the null hypothesis when (x-bar – 11)/(0.6/√ 9) -2.33. Equivalently we reject the null hypothesis when 11 – 2.33(0.2) x-bar, or when x-bar is less than 10.534. We fail to reject the null hypothesis for x-bar greater than or equal to 10.534. If the true population mean is 10.75, then the probability that x-bar is greater than or equal to 10.534 is equivalent to the probability that z is greater than or equal to -0.22. This probability, which is the probability of a type II error, is equal to 0.587.

10 Fascinating Things About Black Widow Spiders

10 Fascinating Things About Black Widow Spiders Black widow spiders are feared for their potent venom, and rightfully so, to some degree. But much of what you think is true about the black widow is probably more myth than fact. Interesting Things About Black Widow Spiders These 10 fascinating facts about black widow spiders will teach you how to identify them, how they behave, and how to minimize your risk of being bitten. Widow spiders arent always black When most people talk about the black widow spider, they likely think theyre referring to a particular spider species. But in the U.S. alone, there are three different kinds of black widows (northern, southern, and western). And although we tend to refer to all members of the genus Lactrodectus as black widows, widow spiders arent always black. There are 31 species of Lactrodectus spiders worldwide. In the U.S., these include a brown widow and a red widow. Only adult female black widows inflict dangerous bites Female widow spiders are larger than males. It is believed, therefore, that female black widows can penetrate vertebrate skin more effectively than males and inject more venom when they bite. Nearly all medically significant black widow bites are inflicted by female spiders. Male widow spiders and spiderlings are rarely a cause for concern, and some experts even say they dont bite. Black widow females rarely eat their mates Lactrodectus spiders are widely thought to practice sexual cannibalism, where the smaller male is sacrificed after mating. In fact, this belief is so widespread the term black widow has become synonymous for femme fatale, a kind of seductress who lures men with the intention of bringing harm to them. But studies show that such behavior is actually quite rare in widow spiders in the wild, and even uncommon among captive spiders. Sexual cannibalism is actually practiced by quite a few insects and spiders and is not unique to the often maligned black widow. Most (but not all) widow spiders can be identified by a red hourglass marking Nearly all black widow females bear a distinct hourglass-shaped marking on the underside of the abdomen. In most species, the hourglass is bright red or orange, in sharp contrast to its shiny black abdomen. The hourglass may be incomplete, with a break in the middle, in certain species like the northern black widow (Lactrodectus variolus). However, the red widow, Lactrodectus bishopi, does not have an hourglass marking, so be mindful that not all widow spiders are identified by this feature. Black widow spiderlings look nothing like the black and red spiders we recognize as black widows Widow spider nymphs are mostly white when they hatch from the egg sac. As they undergo successive molts, the spiderlings gradually darken in color, from tan to gray, usually with white or beige markings. Female spiderlings take longer to reach maturity than their brothers but eventually turn dark black and red. So that drab, pale little spider you found just might be a widow spider, albeit an immature one. Black widows make cobwebs Black widow spiders belong to the spider family Theridiidae, commonly called the cobweb spiders. These spiders, black widows included, construct sticky, irregular silk webs to ensnare their prey. Members of this spider family are also referred to as comb-foot spiders because they have a row of bristles on their back legs to help them wrap silk around their prey. But no need to worry. Although they are closely related to the house spiders building cobwebs in the corners of your home, black widows rarely come indoors. Female black widows have poor eyesight Black widows rely on their silk webs to see whats going on around them because they cant see very well. The black widow female usually hides in a hole or crevice and builds her web as an extension of her hiding spot. From the safety of her retreat, she can feel the vibrations of her web when either prey or predator comes in contact with the silk threads. Male widow spiders looking for mates use this to their advantage. The male black widow will cut and rearrange the females web, making it difficult for her to sense whats happening, before carefully approaching her to mate. Black widow venom is 15 times as toxic as that of the prairie rattlesnake Widow spiders do pack a powerful punch of neurotoxins in their venom. By volume, Lactrodectus venom is an extremely toxic mix of poisons capable of causing muscle cramps, severe pain, hypertension, weakness and sweating in bite victims. But black widow spiders are significantly smaller than rattlesnakes, and theyre built for subduing other small invertebrates, not large mammals like people. When a black widow spider bites a person, the volume of neurotoxins injected in the victim is small.   Black widow spider bites are rarely fatal Although black widow bites can be painful and require medical treatment, they are very rarely fatal.   In fact, the majority of black widow bites cause only mild symptoms, and many bite victims dont even realize they were bitten. In a review of over 23,000 documented Lactrodectus envenomation cases that occurred in the U.S. from 2000 to 2008, the study authors noted that not a single death occurred as a result of a black widow bite. Only 1.4% of bite victims suffered major effects of black widow venom. Before the invention of indoor plumbing, most black widow bites occurred in outhouses Black widows dont often invade homes, but they do like to inhabit human-built structures like sheds, barns, and outhouses. And unfortunately for those who lived before the water closet was commonplace, black widows like to retreat under the seats of outdoor privies, perhaps because the smell attracts so many delicious flies for them to catch. Men who use pit toilets should be aware of this disturbing little factoid – most black widow bites are inflicted on penises, thanks to their tendency to dangle threateningly into the black widows territory beneath the seat. A 1944 case study published in the Annals of Surgery noted that, of 24 black widow bite cases reviewed, eleven bites were on the penis, one was on the scrotum, and four were on the buttocks. A full 16 of the 24 victims were bitten while sitting on the toilet. Sources Physicians Guide to Arthropods of Medical Significance, 6th edition, by Jerome Stoddard.Bugs Rule! An Introduction to the World of Insects, by Whitey Cranshaw and Richard Redak.The Black Widow Spider, by Karen M. Vail, Carl Jones, and Harry Williams, University of Tennessee. Accessed online August 12, 2015.Black Widow Spider, Occupational Safety and Health Administration fact sheet, U.S. Department of Labor. Accessed online August 12, 2015.Black Widow Spider, North Carolina State University. Accessed online August 12, 2015.Black Widow and Other Widow Spiders, University fo California IPM Program. Accessed online August 12, 2015.The Black Widow, Alabama Cooperative Extension System. Accessed online August 12, 2015.Genus Lactrodectus – Widow Spiders, Bugguide.net. Accessed online August 12, 2015.The Treatment of Black Widow Spider Envenomation with Antivenin Latrodectus Mactans: A Case Series, by S. R. Offerman, G. P. Daubert, and R. F. Clark. The Permanente Journal,  15(3), 7 6–81 (2011). Accessed online August 12, 2015. A US Perspective of SymptomaticLatrodectus  spp. Envenomation and Treatment: A National Poison Data System Review, by Andrew A. Monte, Becki Bucher-Bartelson, and Kennon J. Heard. Annals of Pharmacotherapy, 45(12), 1491-1498 (December 2011). Accessed online August 12, 2015.Black Widow Spider Bite, by H. T. Kirby-Smith.  Annals of Surgery, 115(2), 249–257 (1942).