The researchers are concerned that the dollars spent per shopper examined in the study is too different from what has been found in other studies. They expected to see an average spending of 84.5 dollars. Based on our data, is there evidence that the true mean spending for the population is significantly different than the expected average of 84.5 at a 5% significance level? Once again, assume the population standard deviation is 16. a) Perform a hypothesis test BY HAND, completing all the following steps. i. State your hypotheses. iiCalculate the test statistic BY HAND. Round to 2 decimal places and show your work. iii. Give the probability statement for the test statistic and the p-value. Round to 4 decimal places and show your work. b) Draw your conclusions from the test. c) Based on part b), decide whether the researchers' concerns are warranted. In other words, is there a reason to believe that the dollars spent per visit to the mall is different than previously expected?65.51
82.75
69.47
95.71
76.53
67.27
77.3
77.21
66.44
41.97
74.25
93.66
42.68
81.66
73.51
64.1
77.04
102.3
109.06
63.58
98.99
86.7
53.3
90.16
54.09
84.18
88.94
77.88
75.67
74.3
68.3
45.77
44.67
130.84
79.74
81.56
80.12
73.07
85.07
81.45
51.41
67.5
79.27
103.48
37.52
99.72
55.1
88.31
74.23
96.82
61.06
65.92
71.83
62.98
61.04
105.58
75.93
101.92
103.68
90.25
105.94
109.36
76.78
116.37
106.04
60.1
85.64
80.28
71.54
65.76
96.39
91.49
101.92
99.79
86.52
50.82
90.6
84.61
102.25
111.32
52.15
100.33
71.9
89.94
109.53
51.09
101.86
100.82
81.36
74.32
50.45
103.48
49.03
100.68
83.56
109
95.9
76.36
118.13
87.75
120.54
111.93
38.6
76.5
72.12
87.44
85.91
57.4
94.8
50.01
82.3
78.14
75.61
75.32
71.32
103.65
89.97
90.79
86.29
61.31
n=120

To study if the average dollars spend per shopper is different than the expected population average of $84.5 there was a sample of 120 shoppers taken and their spending habit registered.

Then the study variable is:

X: Amount of money spent by a shopper ($)

This population standard deviation is σ= $16

Using the data information I've run a normality test, with p-value: 0.3056 against the test significance level α: 0.05, the decision is to not reject the null hypothesis so there is enough statistical evidence to conclude that the study variable has a normal distribution.

X~N(μ;σ²)

I've also calculated the sample mean and standard deviation:

X[bar]= $81.01

S= $19.62

a.

If the objective is to test that the true mean spending for the population is significantly different than the expected average of 84.5, the parameter of interest is the population mean μ and the hypothesis test will be two-tailed.

H₀: μ = 84.5

H₁: μ ≠ 84.5

α:0.05

The statistic is a standard normal for one sample:

[tex]Z= \frac{X[bar]-Mu}{\frac{Sigma}{\sqrt{n} } } ~N(0;1)[/tex]

[tex]Z_{H_0}= \frac{81.01-84.5}{\frac{16}{\sqrt{120} } } = -2.389[/tex]

p-value two tailed 0.016894

b.

The decision rule using the p-value method is:

If p-value ≤ α ⇒ Reject the null hypothesis.

If p-value > α ⇒ Do not reject the null hypothesis.

Since the p-value is less than the significance level, the decision is to reject the null hypothesis.

c.

Using a level of significance of 5% the null hypothesis was rejected. So there is enough statistical evidence to support the researcher's claim that the true mean spending for the population is significantly different than the expected average of $84.5.

I hope it helps!