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FOR REFERENCES:

O a Let Us Study Before we move forward to the different test statistics, it is important to dene the following terms: o A population includes all of the elements from a set of data. 0 A sample consists of one or more observations drawn from the population. 0 Sample mean (f) is the mean of sample values collected. 0 Population mean (u) is the mean of all the values in the population. If the sample is randomly selected and sample size is large, then the sample mean would be a good estimate of the population mean. 0 Population standard deviation (0) is a parameter which is a measure of variability with xed value calculated from every individual in the population. 0 Sample standard deviation (5) is a statistic which means that this measure of variability is calculated from only some of the individuals in a population. . Population variance (02), in the same sense, indicates how the population data points are spread out. It is the average of the distances from each data point in the population to the mean, squared. Since we already dened important things in identifying the test statistics in hypothesis testing, let us now determine those concepts when given a problem. Let's use the example in Activity 2. Example: A Grade 1 1 researcher reported that the average allowance of Senior High School students was P100. A sample of 40 students has mean allowance of 19120. At a=0.01 test, it was the claimed that the students had allowance of more than P 100.The standard deviation of the population is P50. p = F100 the average allowance of the population (Senior High School students) n=40 the number of students taken from all Senior High School students 9? : P120 the mean allowance ofthe sample 0' = ?50 the standard deviation of the population Now you already know how to get the data needed in choosing test statistics. This time, you will determine what test statistic is appropriate in computing test value in the hypothesis testing. A test statistic is a random variable that is calculated from sample data and used in a hypothesis test. You can use test statistics to determine whether to reject or accept the null hypothesis. The test statistic compares your data with what is expected under the null hypothesis. To identify the test statistic, you must consider whether the population standard deviation / variance is known or unknown. If the population standard deviation 0 is known, then the mean has a normal distribution. Use z-test. If the population standard deviation ois unknown, then the mean has a t distribution. Use t-test. Instead of the population standard deviation, use the sample standard deviation. z-test In a ztest, the sample is assumed to be normally distributed. A z score is calculated with population parameters such as \"population mean\" and \"population standard deviation\". It is used to validate a hypothesis that the sample drawn belongs to the same population. When the variance is known and either the distribution is normal or sample size is large, use a ztest statistic. t-test Like a ztest, a ttest also assumes a normal distribution of the sample. A ttest is used when the population variance or standard deviation are not known. When the variance is unknown and a sample size is less than 30, use a ttest statistic assuming that the population is normal or approximately normal. Central Limit Theorem In Central Limit Theorem, if the population is normally distributed or the sample size is large and the true population mean u = no , then 2 has a standard normal distribution. When population standard deviation ois not known, we may still use zscore by replacing the population standard deviation 0 by its estimate, sample standard deviation s. Since the sample is large the resulting test statistic still has a distribution that is approximately standard normal. Historically, this was very useful, as most statisticians before did not have access to the ttable of quantities for very large number of degrees of freedom. But with modern computers today, using ttest with a very large sample size is not a problem at all. However, since you will be using a ttable with only limited number of degrees of freedom, you will use ztest when the sample size is large even though the population standard deviation is unknown. When sample sizes are small, the Central Limit Theorem does not apply. You must then impose stricter assumptions on the population to give statistical validity to the test procedure. One common assumption is that the population from which the sample is taken has a normal probability distribution to begin with. Under such circumstances, if the population x-no 0' Ni standard deviation is known, then the test statistic still has the standard normal distribution. The table shows what test statistic is appropriate when:| Population Variance Is Population Variance Is Central Limit Theorem Known Unknown (CLT) . . Population is normal or Population may not be Population 1s normally . . . . nearly normally normally dlstrlbuted. dlstrlbuted. . . dlstributed. n 2 30 or considered n 2 30 sufciently large Population standard Sample standard . . . . . . . . Variance IS known/ dev1at10n (0) 1s known. dev1at10n (3) IS known. unknown Population standard deviation (0) is unknown. Use z-test by replacing population standard z-test deviation (0') by sample standard deviation (5) in the formula. Identifying Appropriate Test Statistic When the value of sample size (n)... m oisknown oisnotknown Illustrative Examples: 1. A manufacturer claimed that the average life of batteries used in their electronic games is 150 hours. It is known that the standard deviation of this type of battery is 20 hours. A consumer Wished to test the manufacturer's claim and accordingly tested 100 electronic games using the battery. It was found out that the mean is equal to 144 hours. Here, the sample size (n) is 100 (extremely large) and population standard deviation (20 hours) is known, then the appropriate test statistic to be used is z-test. 2. An English teacher wanted to test whether the mean reading speed of students is 550 words per minute. A sample of 12 students revealed a sample mean of 540 words per minute with a standard deviation of 5 words per minute. At 0.05 signicance level, is the reading speed different from 550 words per minute? The sample size (n) is 12 which is less than 30 and sample standard deviation (5 words per minute) was given. Therefore, the appropriate test is t-test. 3. A study was conducted to look at the average time students exercise. A researcher claimed that in average, students exercise less than 15 hours per month. In a random sample size n= 115, it was found that the mean time students exercise is 33:11.3 hours per month with s = 6.43 hours per month. Since n=1 15, the sample size is large and variance is unknown. Hence, z-test is the appropriate tool. (Central Limit Theorem) Note: The illustrative examples above used standard deviations instead of variances. Variance is the square of the standard deviation and conversely, the standard deviation is the square root of the variance. Hence, if the standard deviation is known in the problem, then basically, variance is also known.|