The value given by a z-table using a z-score of -2.36 is 0.0091, which, when doubled, is 0.0182 or approximately 0.02. Since the alternative hypothesis is not specific about the population mean being either greater than or less than the value in the null hypothesis, we have to consider both tails of the distribution, but by symmetry of the standard normal distribution, we can accomplish this by simply doubling the value we get from using our obtained z-score with a z-table. Taking the difference between our sample mean and the population mean and dividing it by the standard error gives us our z-score (number of standard errors our sample mean is away from the population mean), which is approximately (7.36 - 7.4) / 0.01697 or -2.36. In our example, the standard error of the mean therefore has a value of 0.12 / 50^0.5, or approximately 0.01697. I say would, because unfortunately, we don’t always know the population standard deviation, and so (as it seems they did here, despite knowing the population standard deviation), we are using the sample standard deviation in its place to find an estimate of the standard deviation for the sampling distribution of the sample mean, which is also known as the standard error of the mean. That would give us a standard deviation for the sampling distribution of the sample mean. Since the answer to what we are asking comes from the sampling distribution of the sample mean, we would find the appropriate standard deviation to use by dividing the population standard deviation by the square root of the sample size (since the variance of the sampling distribution is the population variance divided by the sample size, and the standard deviation is the square root of the variance). To use a z-table, we'll need to find the appropriate z-score first. (We could also use a t-table, but it is allowable to just use a z table since our sample size is larger than 30) So now, we can use the normal cumulative density function or a z-table to find this probability. Under that assumption, and noting also that we are given that the population is normally distributed (or that we took a sample size of at least 30 ), we can treat the sampling distribution of the sample mean as a normal distribution. The p-value is the probability of a statistic at least as deviant as ours occurring under the assumption that the null hypothesis is true.
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