A monthly income investment scheme exists that promises variable monthly returns. An investor will invest in it only if he is assured of an average GHS 180 monthly income. He has a sample of 300 months’ returns which has a mean of GHS 190 and a standard deviation of GHS 75. Should he or she invest in this scheme at 5 percent significance level? 

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This is a fairly straightforward statistical test of a sample mean. Knowing the sample mean and standard deviation, you want to find if the population mean will be at least 180, the specified value. Fortunately, with the information given, there is a straightforward way to find this information.

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This is a fairly straightforward statistical test of a sample mean. Knowing the sample mean and standard deviation, you want to find if the population mean will be at least 180, the specified value. Fortunately, with the information given, there is a straightforward way to find this information.

You are testing whether the mean is greater than or equal to a certain value—which is 180—and if this is statistically significant. You would use a z-test for an unknown mean to perform this action. The first thing you need to do with this is to calculate the test statistic, which you can do from the information given in the question.

The equation for the test statistic is the following: Xbar - U0 / (sigma / sqrt(n)) where the variables are the following pieces of data:

Xbar = the sample mean, in this case 190

U0 = the hypothetical population mean, in this case 180

Sigma = the sample standard deviation

n = the number of samples

You will be testing if U (the true population mean) is greater than or equal to U0, so you're hypothesis is U >= U0 at the 5% confidence level.

So, plugging into that equation, you get 190 - 180 / (75 / sqrt(300)), which results in a value or 2.309, which is your z-statistic. This means that the values you have here are 2.309 standard deviations from the mean, in the positive direction. If you translate that into a cumulative percentage for a one-sided hypothesis test (meaning you only care about the upper half of the value since it is greater than the mean), you end up with a 98.95% value, meaning that there is a 98.95% chance that the true mean is not below 180. Since the difference between that number and 100 is less than 5% (1.05%), you can confirm that it is statistically significant at the 5% confidence level, and you can trust that the investment will guarantee you an average of over 180 GHS.

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