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Homework answers / question archive / Adjust the settings in the applet as follows
Adjust the settings in the applet as follows.
"Distribution" to “Normal”,
"Sample size" to “100”,
"Mean" to “50”,
"Std. dev." to “10”,
"Null mean" to “50” and
"Alternative" to “<”.
Then click the “Update applet” button.
a. With the above settings, what is the alternative hypothesis for this hypothesis test being run? Fail to reject the null hypothesis
b. Without any calculations, how do you know that a sample mean of ?¯=50.121 must result in the outcome of "fail to reject the null hypothesis" for the above settings? As the null mean is 50 and the alternative setting is <, we would expect the mean to be less than 50. However, this is not the case with a sample mean of 50.121.
c. Keeping in mind the above settings, suppose only one of the following sample means will result in the outcome of "reject the null hypothesis". Explain which one and how you can draw that conclusion. (Note that no level of significance has been set, so this is really meant to be a conceptual question and not a computational one.)
?¯1=49.7, ?¯2=48.3, ?¯3=51.1, ?¯4=47.8
A sample mean of 47.8 would lead to rejecting the null hypothesis. This is because 47.8 is lower than the population mean of 50, and in the alternative hypothesis, it is expected that the sample mean is significantly lower than the mean.
d. With the settings above, the true population mean is 50 and the null hypothesis is ?0:?≥50 with an alternative hypothesis of ??:?<50. Why would an outcome of "reject the null hypothesis" be considered an error? Specifically which type of error (Type 1 or Type 2) is it?
Null Hypothesis (H0): The true population mean (μ) is greater than or equal to 50. Alternative Hypothesis (Ha): The true population mean (μ) is less than 50. If a test result leads to the conclusion of "rejecting the null hypothesis" when, in reality, the true population mean is 50 or greater, it would be classified as a Type I error. This is because it would mean that there is insufficient evidence to reject the null hypothesis that is actually true.
3. Keep the settings in the applet as follows.
"Distribution" to “Normal”,
"Sample size" to “100”,
"Mean" to “50”,
"Std. dev." to “10”,
"Null mean" to “50” and
"Alternative" to “<”.
Then click the “Update applet” button.
a. If not already set by default, set “Level” to "0.05". What is the corresponding “Critical value”? -1.66
b. Make sure that the "T-statistic" option just above the frequency graph is selected and run 1000 simulations of the hypothesis test by clicking the “1000 tests” button. You should see a histogram distribution of 1000 sample means.
1. Describe the shape of the distribution. symmetric, bell shaped.
2. In which tail of the distribution should the outcomes of "reject the null hypothesis" displayed? left
c. According to the results in the table, how many outcomes are “Reject null”? 53 What proportion of the 1000 tests is this? 0.053 How does this relate to the value of the “Level” (0.05)? When the level of significance is set to .05, it means that there is only a 5 percent chance that the value of a statistic could be obtained as a result of random error. This threshold is often used in statistical hypothesis testing to determine whether a result is statistically significant or not.
d. Click on the left most bar in the distribution, and select one of the runs represented in that bar. (There may only be one run, but if there are multiple runs, choose which ever one that you like.)
1. What is the “Mean” for that sample? 51.309
2. What is the “T-statistic” for the sample? 1.408
3. What is the P-value for the sample? 0.919
e. How does the T-statistic for this sample run compare to the “Critical value” for the test? When conducting a t-test, we compare the calculated t-statistic to the critical value of t. If the absolute value of the calculated t-statistic is greater than the critical value of t, we reject the null hypothesis
