Identifying Problems When Obtaining Population Parameters

Identifying Problems When Obtaining Population ParametersWe estimate people parameters, such as the call up, base on the render statistics. It is difficult to add a precise apprise or point estimation of these figures. A more practical and informative get is to find a range of cheers in which we expect the population parameters will fall. Such a range of values is called a potency detachment.1. self-reliance INTERVAL renderingThe confidence interval is a range of values constructed from exemplification data so that the population parameter is likely to occur deep down that range at a specified probability. The specified probability is called the take aim of confidence.The shape of the probability dispersion of the seek besotted allows us to specify an interval of specific probability that the population destine, , will fall into.1.1 Large Sample Or quantity Deviation Is KnownCase 1The tired passing is known orIt is a large adjudicate (i.e. at least 30 observati ons).The Central Limit Theorem states that the sampling distribution of the savor means is some recipe. We can use the tables in the Appendix to find the appropriate Z value.Key PointsThe exemplification normal distribution allows us to draw the following conclusions68% of the sample means will be within 1 stock(a) departures of the population mean, .95% of the sample means will be within 1.96 measuring deviations of the population mean, .99% of the sample means will deceitfulness within 2.58 step deviations of the population mean.These intervals are called the confidence interval.The standard deviation supra (i.e. the standard delusion) is referring to the standard deviation of the sampling distribution of the sample mean.Locating 0.475 in the body of the table, read the alike row and column values, the value is 1.96. Thus, the probability of purpose a Z value mingled with 0 and 1.96 is 0.475. Likewise, the probability of being in the interval between -1.96 and 0 is also 0.475. When we combine these two, the probability of being in the interval of -1.96 to 1.96 is thereof 0.95.1.1.1 How do you compute a 95% confidence interval?Assume our research involves the annual starting salary of short letter graduates in a local university. The sample mean is $39,000, while the standard deviation of the sample mean is $250. Assume our sample contains more than 30 observations. The 95% confidence interval is between $38,510 and $39,490. Found by $39,000 +/- 1.96($250)In most situations, the population standard deviation is non available, so we estimate it as follows (Standard Error)Conclusions95% confidence interval99% confidence intervalConfidence interval for the population mean (n 30)Z depends on confidence take aimExample 1The Hong Kong Tourist connexion wishes to have information on the mean annual income of tour guides. A random sample of 150 tour guides reveals a sample mean of $45,420. The standard deviation of this sample is $2,050. The asso ciation would like answers to the following questions(a) What is the population mean?The best estimate of the unknown population value is the corresponding sample statistic. The sample mean of $45,420 is a point estimate of the unknown population mean.(b) What is a reasonable range of values for population mean?The Association decides to use the 95% level of confidence. To determine the corresponding confidence interval, we use the formulaThe endpoints would be $45,169 and $45,671 and they are called confidence limits. We could expect about 95% of these confidence intervals contain the population mean. About 5% of the intervals would not contain the population mean annual income, i.e. the .Figure 2 Probability distribution of population mean1.2 Small Sample Or Standard Deviation Is UnknownCase 2The sample is small (i.e. less than 30 observations) or,the population standard deviation is not known.The correct statistical procedure is to replace the standard normal distribution with th e t distribution. The t distribution is a continuous distribution with numerous similarities to the standard normal distribution.1.2.1 Standard normal distribution versus t distributionFigure 3 Z distribution versus t distributionThe t distribution is flatter and more spread out than the standard normal distribution.The standard deviation of the t distribution is larger than the normal distribution.Confidence interval for a sample with unknown population mean, . The confidence interval isAssume the sample is from a normal population.Estimate the population standard deviation () with the sample standard deviation (s).Use t distribution rather than the Z distribution.Example 2A shoe maker wants to investigate the useful life of his products. A sample of 10 pairs of shoes that had been walked for 50,000 km showed a sample mean of 0.32 inch of sole remaining with a standard deviation of 0.09 cm. Constructing a 95% confidence interval for the population mean, would it be reasonable for the manufacturer to conclude that after 50,000 km the population mean amount of sole remaining is 0.3 cm?Assume the population distribution is normal. The sample standard deviation is 0.09 cm.There are only 10 observations and hence, we use t distributionEstimation= 0.32, s = 0.09, and n = 10.Step 1 Locate t by moving across the row for the level of confidence compulsory (i.e. 95%).Step 2 The column on the left margin is identified as df. This refers to the anatomy of degrees of freedom. The spot of degree of freedom is the number of observations in the sample minus the number of samples, written n-1.(i.e. 10-1=9).Step 3 Confidence Interval =The endpoints of the confidence interval are 0.256 and 0.384.Step 4 recitation the manufacturer can be evenhandedly sure (95% confident) that the mean remaining tread depth is between 0.256 and 0.384 cm. Because 0.3 is in this interval, it is possible that the mean of the population is 0.3.2. CHOOSING AN APPROPRIATE SAMPLE SIZEThe necessar y sample size depends on three factorsLevel of confidence wanted To increase level of confidence, increase n.Margin of error the researcher will tolerate To overturn allowable error, increase n.Variability in the population being studied For a more widely dispersed sample, increase n.We can express the fundamental interaction among these three factors and the sample size in the following formulaSample size for estimating the population mean,Noten Sample sizeZ Standard normal valueS Estimate of population standard deviationE Maximum allowable errorExample 3An accounting assimilator wants to know the mean amount that independent directors of small companies earn per month as remuneration for being a director. The error in estimating the mean is to be less than $100 with a 95% level of confidence. The student plant a report by the government that estimated the standard deviation to be $1000. What is the required sample size?Maximum allowable error, E, is $100.Value of Z for a 95% le vel of confidence is 1.96, and the estimate of the standard deviation is $1000.Substitute into , we getn = (1.96) (1000) 2 = 19.62 = 384.16100The sample of 385 is required to meet the requirements. If the students want to increase the level of confidence, e.g. 99%, this requires a larger sample.Z = 2.58, son = (2.58) (1000) 2 = 25.82 = 665.64100Sample = 6663. WHAT IS A HYPOTHESIS?Definitions shot is a statement about a population parameter developed for the purpose of exam.Hypothesis testing is a procedure based on sample evidence and probability theory to determine whether the supposal is a reasonable statement.In statistical analysis, we always make a claim about the population parameters, i.e. a shot. We collect data and then use the data to test the assertion.4.1 Five-Step Procedure For testing A HypothesisFigure 4 How to test a scheme4.1.1 Step 1 utter unserviceable theory (H0) and alternative dead reckoning (H1)The commencement exercise step is to state the hypothesis being tested. It is called the null hypothesis. We either reject or fail to reject the null hypothesis. Failing to reject the null hypothesis does not establish that H0 is true.The null hypothesis is a statement that is not rejected unless our sample data provide convincing evidence that it is false.The alternative hypothesis is a statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.Example 4A journal has tell that the mean age of commercial helicopters is 15 years. A statistical test of this statement would first need to determine the null and the alternate hypotheses.The null hypothesis represents the current or reported condition. It is written H0 = 15.The alternate hypothesis is that the statement is not true, i.e. H1 15.4.1.2 Step 2 Select a level of momentThe level of substance is the probability of rejecting the null hypothesis when it is true.A ratiocination is do to use the 5% level, 1% level, 10% level or any former(a) level between 0 and 1. We must decide on the level of significance before formulating a finis rule and collecting sample data. fiber I error Rejecting the null hypothesis, H0, when it is true.Type II error Accepting the null hypothesis when it is false.Example 5Suppose AA accompany Ltd has informed bracelet suppliers to bid for learn on the supply of a large amount of bracelets. Suppliers with the lowest bid will be awarded a sizable contract.Suppose the contract specifies that the watch producers quality-assurance department will take samples of the communique.H0 The shipment of bracelet contains 6% or less substandard bracelets.H1 More than 6% of the boards are defective.A sample of 50 bracelets received August 2 from BB Metals Ltd revealed that four bracelets, or 8%, were substandard. The shipment was rejected because it exceeded the maximum of 6% substandard bracelets. If the shipment was actually substandard, the decision to return the bracelets to the supplier was correct.However, suppose the four substandard bracelets selected in the sample of 50 were the only substandard bracelets in the shipment of 4,000 bracelets. Then only 1/10 of 1% were defective (4/4000 = 0.001). In that case, less than 6% of the entire shipment was substandard and rejecting the shipment was an error.We may have rejected the null hypothesis that the shipment was not substandard when we should have accepted the null hypothesis.By rejecting a true null hypothesis, we committed a Type I error.AA Watch Ltd would commit a Type II error if, unknown to the company an incoming shipment of bracelet from BB Metals Ltd contained 15% substandard bracelets, yet the shipment was accepted. How could this meet?Suppose two out of the 50 bracelets in the sample (4%) tested were substandard, and 48 out of the 50 were good bracelets. As the sample contained less than 6% substandard bracelets, the shipment was accepted but it could be purely by chance that the 48 good bracelets selec ted in the sample were the only acceptable ones in the entire shipment.In conclusionNull HypothesisAccepts H0Rejects H0H0 is trueCorrect decisionType I errorH0 is falseType II errorCorrect decision4.1.3 Step 3 Select the test statisticsThere are many test statistics. In this chapter, we use both Z and t as the test statistic.DefinitionA test statistic is a value, determined from sample information, used to determine whether to reject the null hypothesis.In hypothesis testing for the mean () when is known or the sample size is large, the test statistic Z is computed byThe Z value is based on the sampling distribution of , which follows the normal distribution when the sample is reasonably large with a mean () equal to , and a standard deviation , which is equal to . We can thus determine whether the deflection between and is statistically significant by finding the number of standard deviations is from , using the formula above.4.1.4 Step 4 Formulate the decision ruleDefinitionA d ecision rule is a statement of the specific conditions under which the null hypothesis is rejected and the conditions under which it is not rejected.The region or flying field of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote.The theatre of operations where the null hypothesis is not rejected is to the left of 1.65.The area of rejection is to the right of 1.65.A one-tailed test is being applied.The 0.05 level of significance was chosen.The sampling distribution of the statistic Z is normally distributed.The value 1.65 separates the regions where the null hypothesis is rejected and where it is not rejected.The value 1.65 is the critical value.The critical value is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.Figure 5 Area of rejection for the null hypothesis4.1.5 Step 5 Make a decisionThe fina l step in hypothesis testing is computing the test statistic, comparing it to the critical value, and making a decision to reject or not to reject the null hypothesis.Based on the information, Z is computed to be 2.34, the null hypothesis is rejected at the 0.05 level of significance. The decision to reject H0 was made because 2.34 lies in the region of rejection, i.e. beyond 1.65.We would reject the null hypothesis, reasoning that it is highly improbable that a computed Z value this large is due to sampling variation.Had the computed value been 1.65 or less, say 0.71, the null hypothesis would not be rejected. It would be reasoned that such a small computed value could be attributed to chance.Example 6A large car leasing company wants to buy tires that average about 60,000 km of wear under normal usage. The company will, therefore, reject a shipment of tires if tests reveal that the life of the tires is significantly below 60,000 km on the average.The company would be glad to acce pt a shipment if the mean life is great than 60,000 km. However, it is more concerned that it will have sample evidence to conclude that the tires will average less than 60,000 km of useful life. Thus, the test is set up to satisfy the concern of the car leasers that the mean life of the tires is less than 60,000 km.The null and alternate hypotheses in this case are written H0 60,000 and H1 In this problem, the rejection region is pointing to the left, and is therefore in the left tail.SummaryIf H1 states a direction, we use a one-tailed test.If no direction is specified in the alternate hypothesis, we use a two-tailed test.Figure 6 One-tailed test5. TESTING FOR POPULATION MEAN WITH KNOWN POPULATION STANDARD DEVIATION5.1 Two-tailed TestABC Watch Ltd manufactures luxury watches at several plants in Europe. The weekly output of the Model A33 watch at the Swiss Plant is normally distributed, with a mean of 200 and a standard deviation of 16. Repennyly, because of securities indust ry expansion, mechanisation has been introduced and employees laid off. The chief executive officer would like to investigate whether there has been a change in the weekly takings of the Model A33 watch. To put it an another(prenominal) way, is the mean output at Swiss Plant different from 200 at the 0.01 significant levels?5.1.1 Step 1 State null hypothesis and alternate hypothesisThe null hypothesis is The population mean is 200. H0 = 200.The alternate hypothesis is The mean is different from 200. H1 200.5.1.2 Step 2 Select the level of significanceThe 0.01 level of significance is used. This is , the probability of committing a Type I error, and it is the probability of rejecting a true null hypothesis.5.1.3 Step 3 Select the test statisticThe test statistic for the mean of a large sample is Z.Figure 7 Normalise the standard deviation5.1.4 Step 4 Formulate the decision ruleThe decision rule is speculate by finding the critical values of Z from Appendix D.Since this is a two -tailed test, half of 0.01, or 0.005, is placed in each tail. The area where H0 is not rejected, i.e. area between the two tails, is 0.99.Appendix D is based on half of the area under the curve, or 0.5. Then 0.5 0.005 is 0.495, so 0.495 is the area between 0 and the critical value.The value nearest to 0.495 is 0.4951. Then read the critical value in the row and column corresponding to 0.4951. It is 2.58.Decision ruleReject H0 if the computed Z value is not between -2.58 and +2.58.Do not reject H0 if Z falls between -2.58 and +2.58.Figure 8 Two-tailed test5.1.5 Make a decision and interpret the resultCompute Z and apply the decision rule to decide whether to reject H0.The mean number of watches produced weekly for last year is 203.5. The standard deviation of the population is 16 watches.Because 1.55 does not fall in the rejection region, H0 is not rejected. We conclude that the population mean is not different from 200.So we would report to the CEO that the sample evidence does not show that the production rate at the Swiss plant has changed from 200 per week. The difference of 3.5 units between the historical weekly production rate and the mean number of watches produced weekly for last year can reasonably be attributed to sampling error.Figure 9 Rejection regions for the two-tailed testSo did we prove that production rate is still 200 per week?No Failing to disprove the hypothesis that the population mean is 200 is not the same affair as proving it to be true.5.2 P-value In Hypothesis TestingDefinitionP-value is the probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true.How confident are we in rejecting the null hypothesis?This approach reports the probability of getting a value of the test statistic at least as extreme as the value actually obtained. This process compares the probability called the P-value, with the significant level.If the P-value If the P-value significant level, H0 is not rejected.A actually small P-value, such as 0.0001, indicates that there is little likelihood the H0 is true. If a P-value of 0.2033 means that H0 is not rejected, there is little likelihood that it is false.Figure 10 P-valueP-valueInterpretationLess than 0.1Some evidence that H0 is not trueLess than 0.05Strong evidence that H0 is not trueLess than 0.01Very strong evidence that H0 is not trueLess than 0.001Extremely strong evidence that H0 is not trueThe probability of finding a Z value of 1.55 or more is 0.0606, found by 0.5 0.4394.The probability of obtaining an greater than 203.5 if = 200 is 0.0606.To compute the P-value, we need to be concerned with the region less than -1.55 as well as the values greater than 1.55. The two-tailed P-value is 0.1212, found by 2(0.0606). The P-value of 0.1212 is greater than the significance level of 0.01, so H0 is not rejected.Chapter ReviewThe Central Limit Theorem states that the sampling distribution of the sample means is slightly normal.The standard error refers to the standard deviation of the sampling distribution of the sample mean.We use t distribution when the sample is less than 30 observations and the population standard deviation is not known.The necessary sample size depends on 1) level of confidence wanted 2) margin of error the researcher will tolerate 3)variability in the population.By rejecting a true null hypothesis, we committed a Type I error.We would reject the null hypothesis when it is highly improbable that a computed Z value this large is due to sampling variation.What You Need To KnowConfidence interval A range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability.Hypothesis A statement about a population parameter developed for the purpose of testing.Hypothesis testing A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.Critic al value The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.P-value The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true.Work Them Out1. The average number of eld in outdoors assignments per year for salespeople employed by an electronic wholesaler needs to be estimated with a 0.90 degree of confidence. In a small sample, the mean was 150 days and the standard deviation was 14 days. If the population mean is estimated within two days, how many salespeople should be interviewed?A 134B 152C 111D 1202. A random sample of 85 staff of managerial grade revealed that a person spent an average of 6.5 years on the romp before being promoted. The standard deviation of the sample was 1.7 years. Using the 0.95 degree of confidence, what is the confidence interval for the population mean?A 6.19 and 6.99B 6.15 and 7.15C 6.14 and 6.86D 6.19 and 7 .193. The mean weight of lorries travelling on a particular highway is not known. A state highway authority needs an estimate of the mean. A random sample of 49 lorries was selected and finds the mean is 15.8 tons, with a standard deviation of 3.8 tons. What is the 95 per cent interval for the population mean?A 14.7 and 16.9B 14.2 and 16.6C 14.0 and 18.0D 16.1 and 18.14. A bank wants to estimate the mean balances owed by platinum Visa card holders. The population standard deviation is estimated to be $300. If a 98% confidence interval is used and an interval of $75 is desired, how many platinum cardholders should be taken into sample?A 84B 82C 62D 875. A sample of 20 is selected from the population. To determine the appropriate critical t-value, what number of degrees of freedom should be used?A 20B 19C 23D 276. If the null hypothesis that two means are equal is true, where will 97% of the computed z-values lie between?A 2.58B 2.38C 2.17D 1.687. Suppose we are testing the differ ence between two proportions at the 0.05 level of significance. If the computed z is -1.57, what is our decision?A Reject the null hypothesisB Do not reject the null hypothesisC Review the sampleD protest judgment8. The net weights of a sample of bottles filled by a machine manufactured by Dame, and the net weights of a sample filled by a similar machine manufactured by Putne Inc, are (in grams)Dame 5, 8, 7, 6, 9 and 7Putne 8, 10, 7, 11, 9, 12, 14 and 9Testing the claim at the 0.05 level that the mean weight of the bottles filled by the Putne machine is greater than the mean weight of the bottles filled by the Dame machine, what is the critical value?A 2.215B 2.175C 1.782D 1.6829. Which of the following conditions must be met to conduct a test for the difference in two sample means?A information must be of interval scaleB Normal distribution for the two populationsC Same variances in the two populationsD All the above are correct10. Take two independent samples from two population s in order to determine if a statistical difference on the mean exists. The number for the first sample and the number in the second sample are 15 and 12 respectively. What is the degree of freedom associated with the critical value?A 24B 25C 26D 27SHORT QUESTIONSA consumer group would like to estimate the mean monthly water ride for a single family house in June within $5 using a 99% level of confidence. Similar research has found that the standard deviation is estimated to be $25.00.What would be the sample size?The manager of the Kingsway Mall wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent.$48 $42 $46 $51 $23 $41 $54 $37 $52 $48$50 $46 $61 $61 $49 $61 $51 $52 $58 $43What is the best estimate of the population mean?Determine a 99 per cent confidence interval. Interpret the result.Would it be reasonable to conclude that the population mean is $50? What about $60?ESSAY QUESTION1. ABC Film Ltd kno ws that a certain favourite movie ran an average of 84 days, and the corresponding standard deviation was 10 days. The manager of New Westminster district was interested in comparing the movies popularity in his region with that in all of Canadas other theatres. He randomly selected 70 theatres in his region and found that they showed the movie for an average of 82 days.(a) State appropriate hypotheses for testing whether there was a significant difference in the length of the pictures run between theatres in the New Westminster district and all of Canadas other theatres.(b) Test these hypotheses at a 1% significance level.