l population h öf the standard error of the mean that is used ratio- A replaceme
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l population h öf the standard error of the mean that is used ratio- A replacement for the z ratio whenever the unknown population s estimated. deviation must be estimated. - the number of values free to vary, given one or a sample of observed values used to estimate some unkno Text Review more mathematical restrictions on wn population characteristic. The sampling distribution of t was discovered by (1) In reality, there is a family of t distribution, each associated with a number referred to as (2) In the present case, the number of degrees of freedom (df) always equals (3) cal, unimodal, and bell of the t All t distributions are similar to z distributions in that they are symmetri shaped. The biggest difference between t and z distributions is the (4) distribution. alues that correspond to the common levels of and originate from the (6) Tables for t distributions contain only the v significance. The t values listed in the table are (5) half of the distribution. The symmetry allows the critical t values for the lower half of the distribution to be obtained by simply placing a negative sign in front of any table entry The t distribution has greater variability than z distribution. This increased variability arises from the estimated standard error of the mean and the chance differences that could occur in the estimates. The extra variability also explains the inflated tails of the curve in the t distribution. Because of this extra variability, one could expect actual critical t values to be (7) values. than critical z All hypothesis tests represent variations on a common theme. If some observed characteristic. such as the mean for a random sample, qualifies as a rare outcome under the null hypothesis, the hypothesis will be (8) To determine whether an outcome is rare, the observed characteristic is converted to a new value, such as t, and compared to critical values from the appropriate sampling distribution. Otherwise, the hypothesis will be (9) To construct confidence intervals for estimating the population mean based on a t distribution, use the formula for confidence intervals previously presented and substitute t for z. The symmetrical limits of a confidence interval require specifications like those of a two-tailed hypothesis test. Use of the t test requires some basic assumptions. Use t rather than z when the (10) is unknown. Assume that the underlying population is (11) If the assumption of normality is violated, the accuracy of the test is relatively is sufficient. fected as long as (12) 24Explanation / Answer
William Sealy Gosset
Student t statistic was discovered by William Sealy Gosset in 1908 when he was a student. He published his discovery under a pseudo name Student. So the name student t distribution.
Cumulative probability
The probability of the values of the random samples fall within the range of 2 or 3 standard deviation. Cumulative probability is the sum of the probabilities under the curve towards upper or lower tail critical region from man.
df=Sample size -1
Degrees of freedom is sample size minus one. One sample in the distribution is mean of the sample. So only n-1 samples are distributed in the graph.
Smaller Sample size n<30 and unknown population standard deviation
Parameter is compared in the student t test statistics if the population means are same or different compared to the earlier parameter. In special case like this test, population standard deviation is unknown. So the sample standard deviation is used to test the means of the population from a sample data of smaller size, less than 30
Critical values
Critical values are the area under the curve from the mean. That spreads from mean to right and to the left.
Upper or Lower
Population mean is increased or same or decreased, the lower than the mean or upper than the mean is the criteria of the student t test statistic
greater
Student t test is created to address the larger variability of the population parameter, critical values in this test is greater than the z critical values under the same condition with lesser sample size and unknown population standard deviation.
rejected
The question of finding an answer if the population mean is within the critical values. If the test t statistic is lesser than lower critical value or greater than the upper critical, we we will reject the null hypothesis.
accepted
Continuing 8, otherwise do not reject null hypothesis.
Populaton standard deviation
Unknown population standard deviation is the main criteria to chose the t test. When we donot know the standard deviation of the entire population, but we can calculate the sample standard deviation from the samll number of samples, student t test will be used.
normally distributed
Population is normally distributed even though the sample size is smaller. It means that the sample size is larger, it will result in normal distribution.
sample size
If the sample is larger than 30, but the samples are not normally distributed, then also the student t test statistic will produce better results.
1William Sealy Gosset
Student t statistic was discovered by William Sealy Gosset in 1908 when he was a student. He published his discovery under a pseudo name Student. So the name student t distribution.
2Cumulative probability
The probability of the values of the random samples fall within the range of 2 or 3 standard deviation. Cumulative probability is the sum of the probabilities under the curve towards upper or lower tail critical region from man.
3df=Sample size -1
Degrees of freedom is sample size minus one. One sample in the distribution is mean of the sample. So only n-1 samples are distributed in the graph.
4Smaller Sample size n<30 and unknown population standard deviation
Parameter is compared in the student t test statistics if the population means are same or different compared to the earlier parameter. In special case like this test, population standard deviation is unknown. So the sample standard deviation is used to test the means of the population from a sample data of smaller size, less than 30
5Critical values
Critical values are the area under the curve from the mean. That spreads from mean to right and to the left.
6Upper or Lower
Population mean is increased or same or decreased, the lower than the mean or upper than the mean is the criteria of the student t test statistic
7greater
Student t test is created to address the larger variability of the population parameter, critical values in this test is greater than the z critical values under the same condition with lesser sample size and unknown population standard deviation.
8rejected
The question of finding an answer if the population mean is within the critical values. If the test t statistic is lesser than lower critical value or greater than the upper critical, we we will reject the null hypothesis.
9accepted
Continuing 8, otherwise do not reject null hypothesis.
10Populaton standard deviation
Unknown population standard deviation is the main criteria to chose the t test. When we donot know the standard deviation of the entire population, but we can calculate the sample standard deviation from the samll number of samples, student t test will be used.
11normally distributed
Population is normally distributed even though the sample size is smaller. It means that the sample size is larger, it will result in normal distribution.
12sample size
If the sample is larger than 30, but the samples are not normally distributed, then also the student t test statistic will produce better results.
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