# One Of The Assumptions We Sometimes Need To Make When Performing Statistical Inferences Is That The Response Variable In The Population Has A Normal Distribution. Is It Possible To Check That This Assumption Is Satisfied?

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One of the assumptions we sometimes need to make when performing statistical inferences is that the response variable in the population has a Normal distribution. Is it possible to check that this assumption is satisfied?

• No – we can’t really check this assumption since all data will look perfectly bell-shaped and symmetric, even if the population was not Normal.
• Yes – we can be absolutely sure the population was Normal if a plot of the data has no major outliers.
• No – we can’t really check this assumption since we don’t have the whole population, but the t distribution is robust to modest departures from Normality, so we can use it if a plot of the data has no major outliers.
• Yes – we can be absolutely sure the population was Normal if a plot of the data looks roughly bell-shaped and symmetric.
• Yes – we can be absolutely sure the population was Normal if a plot of the data looks perfectly bell-shaped and symmetric.

The correct answer is: No – we can’t really check this assumption since we don’t have the whole population, but the t distribution is robust to modest departures from Normality, so we can use it if a plot of the data has no major outliers.

### What is Statistical Inference?

A common assumption made when doing statistical inferences is that the response variable in the population has a normal distribution. This assumption is important because many statistical procedures are based on normal distribution. When the response variable does not have a normal distribution, the results of the statistical procedure may be inaccurate.

### What is a Type I Error?

A type I error is when you conclude that there is a difference between two groups when there actually is no difference. For example, you may conclude that treatment A is better than treatment B when in reality they are both equally effective. The probability of making a type I error is denoted by alpha (α).

### What is a Type II Error?

A type II error is when you conclude that there is no difference between two groups when in reality there is a difference. For example, you may conclude that treatments A and B are equally effective when in reality, treatment A is better than treatment B. The probability of making a type II error is denoted by beta (β).

### What is the Power of a Statistical Test?

The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it’s the likelihood that you won’t make a type II error. The power of a test is affected by several factors, including:

• The size of the difference between the groups being compared (the larger the difference, the easier it will be to detect)
• The variability of the response variable (the more variable the responses are, the harder it will be to detect a difference)

One of the assumptions we sometimes need to make when performing statistical inferences is that the response variable in the population has a Normal distribution. Is it possible to check that this assumption is satisfied?
We can’t be sure that population is normal, since we don’t have the whole population. However, the t distribution is robust to modest departures from Normality, so we can use it if a plot of the data has no major outliers.

A scientist who studies teenage behavior was interested in determining if teenagers spend more time playing computer games then they did in the 1990s. In 1990s, the average amount of time spent playing computer games was 10.2 hours per week. Is the amount of time greater than that for this year? Twenty students were surveyed and asked how many hours they spent playing video games. The test statistics is equal to 1.39. What is the p-value?
0.0903

A restaurant decides to test their oven’s thermostat to see if it is working properly, that is, if the actual temperature inside the oven is the same as the temperature to which the thermostat was set. Twenty times, the oven was set at 350 degrees and then the temperature was measured with a thermometer. Ho: mu = 350 and Ha: mu is not equal to 350 The test statistic is equal to 3.01. What is the p-value?
p-value is between 0.002 and 0.01

For each of the following situations, determine which table should be used for making inferences about the population mean, mu.
t table: small n and no outliers in the sample (so population could be normal) neither the Z nor the t tables are appropriate: small n, non-normal population (there is at least one extreme outlier in the sample) either the t or the Z tables would work: large n, any shape population

Did Americans work less than 40 hours a week on average in 1980? In 1980, the GSS included questions about the number of hours that the respondent worked per week. The average number of hours worked per week was 39.61 hours with a standard deviation of 14.41 hours. A sample of 30 respondents was questioned. Find the test statistic.
-0.148

A restaurant decides to test their oven’s thermostat to see if it is working properly, that is, if the actual temperature inside the oven is the same as the temperature to which the thermostat was set. Twenty times, the oven was set at 350 degrees and then the temperature was measured with a thermometer. The chef wants to know if the average oven temperature is different from 350, when the thermostat is set at 350. What is the correct null and alternative hypothesis for this test?
Ho: mu = 350 Ha: mu does not equal 350

A restaurant decides to test their oven’s thermostat to see if it is working properly, that is, if the actual temperature inside the oven is the same as the temperature to which the thermostat was set. Twenty times, the oven was set at 350 degrees and then the temperature was measured with a thermometer. Ho: mu = 350 and Ha: mu is not equal to 350 The test statistic is equal to 1.02. What is the p-value?
p-value greater than 0.20

Did Americans work less than 40 hours a week on average in 1976? In 1976, the GSS included questions about the number of hours that the respondent worked per week. The average number of hours worked per week was 39.28 hours with a standard deviation of 13.47 hours. A sample of 28 respondents was questioned. Find the test statistic.
-0.28

Did Americans work less than 40 hours a week on average in 1974? In 1974, the GSS included questions about the number of hours that the respondent worked per week. The average number of hours worked per week was 39.70 hours with a standard deviation of 8.88 hours. A sample of 44 respondents was questioned. Find the test statistic.
-0.22