Question

• Select two data values from your raw data – one that is inside of the confidence interval and one that is outside – one must be at the high end of the data and one at the low end – and construct two hypothesis tests, one for each value. One of the tests should be a “less than”, the other should be a “greater than”, depending on the value being tested. Use a 95% level of confidence. Showcase Ho and Ha.
• 59
  58.7
  59.6
  60.1
  63.2
  66
  58.4
  60
  62.9
  63.1
  59.7
  66.2
  63.4
  66
  69
  63.1
  64.8
  66.9
  64.2
  65.8
  63.5
  63.4
  57.4
  54.9
  58.4
  55
  51.7
  57.5
  50.1
  58.9
  57.5
  58.5
  57.3
  67.1
  71.2
  61.9
  63.5
  66
  61.3
  72.5
  67.7
  65
  50
  58.4
  53.3
  52.7
  52
  50.6
  52.6
  51.4
  49.3
  52.6
  51.9
  57.2
  62.1
  61.8
  59.1
  60.6
  61.6
  60.9
  61
  60.8
  65
  64.8
  55
  54.3
  56
  51.2
  56.6
  58.9
  52.9
  50.9
  55.5
  54.5
  56.7
  55
  53.5
  56.5
  54.8
  54.8
  58.9
  56
  56.3
  54.7
  57.2
  57.8
  61
  60.4
  58.3
  58.9
  56.9
  58.9
  57.4
  61.7
  60.6
  62.1
  54.7
  62.7
  59
  59.3
  60
  59.1
  59.5
  54.7
  61.2
  55.8
  57.5
  57.5
  61.9
  63.7
  62.3
  61.5
  59.3
  55.9
  58.8
  56.8
  64.6
  58.5
  58.5
  57.1
  60.5
  58.1
  58.5
  60.7
  60.7
  53.5
  61.7
  57.4
  57
  60.3
  61.2
  56.1
  56
  58.9
  58.8
  63.6
  58.1
  66.5
  59.1
  60.4
  57.1
  56.1
  59.1
  56.4
  71.2
  68.4
  51.9
  67.6
  56.9
  72.9
  53.6
  56.8
  62.1
  57.3
  68.6
  60.8
  50.1
  59.4
  52.5
  61.4
  64
  52.4
  63.8
  64.8
  55.5
  63.3
  74.4
  60.1
  54.7
  50
  57.8
  59.2
  69.2
  50.3
  60
  69.8
  64.7
  69.5
  68.1
  47.2
  54.1
  58
  55.4
  57.3
  67.7
  63.5
  62.8
  61.5
  54.3
  68.5
  60.6
  65.3
  60.2
  61.1
  57.7
  62.3
  57.5
  58.5
  60.2
  55.5
  68.8
  60.6
  63.5
  63.3
  58.9
  60.7
  53
  56
  51.1
  63.7
  56.3
  60
  57.1
  62.5
  53.8
  54.7
  63.9
  53.7
  57.7
  64.2
  59.7
  54.4
  57.4
  50.8
  56.6
  64
  58.4
  55.4
  53
  59.3
  57.1
  56.8
  54.3
  54.4
  59.3
  63.4
  61.2
  59.2
  50.7
  61.3
  63.7
  65.6
  58.2
  54.3
  63.3
  49.6
  59.6
  60.3
  56
  59.3

Answer 1

Since we need to pick 2 values,  one that is inside of the confidence interval and one that is outside, we first need to find the confidence interval.

Using the sample we find the following

n=250 is the sample size

The sample mean is

The sample standard deviation is

We estimate the population standard deviation using the sample

The standard error of mean is

Since the sample size n is greater than 30, we can use normal distribution as the sampling distribution of mean.

The significance level for 95% confidence interval is .

This is the area under both the tails. The area under the right tail is 0.05/2=0.025.

The critical value of z is

Using standard normal table we get that for z=1.96, P(Z<1.96) = 0.975

Hence the critical value is

Now the 95% confidence interval for true mean is

We need to pick one that is inside of the confidence interval and one that is outside.

We will pick 59 as the value which is inside the confidence interval (58.64,59.86), but on the lower side.

We will pick 60 as the value which is outside the confidence interval but on the higher side.

The first is the  “greater than” hypotheses

Let be the true mean of this set of data. We want to test if the true mean value for this population is greater than 59.

(here mention the population. Suppose this sample represents number of miles walked per week by a student, then we have a hypothesis that the true mean number of miles walked per week by a student is greater than 59 miles. Then we use the sample mean which is 59.25 to test if this is true)

The hypotheses are

The hypothesized value of mean (from the null hypothesis) is

The test statistics is

This is a right tailed test (The alternative hypothesis has ">"). The critical value of z for alpha=0.05 is

Using the standard normal tables we get the critical value = 1.645

We will reject the null hypothesis if the test statistics is greater than the critical value.

Here, the test statistics is 0.803 and it is less than the critical value 1.645. Hence we do not reject the null hypothesis.

We conclude that at 5% significance, there is not sufficient evidence that the mean is greater than 59

The second hypotheses is "Less than"

Let be the true mean of this set of data. We want to test if the true mean value for this population is less than 60.

(here mention the population. Suppose this sample represents number of miles walked per week by a student and we have a hypothesis that the true mean number of miles walked per week by a student is less than 60 miles. Then we use the sample mean which is 59.25 to test if this is true)

The hypotheses are

The hypothesized value of mean (from the null hypothesis) is

The test statistics is

This is a left tailed test (The alternative hypothesis has "<"). The critical value of z for alpha=0.05 is

Using the standard normal tables we get the critical value = -1.645

We will reject the null hypothesis if the test statistics is less than the critical value.

Here, the test statistics is -2.415 and it is less than the critical value -1.645. Hence we reject the null hypothesis.

We conclude that at 5% significance, there is sufficient evidence that the mean for the population is less than 60